rewritten in terms of $z$ instead of $x$ (using $z^{2}=(1+x)^{2}/x$). Since
the expansions contain the unknown constants $\tilde{\kappa}_{i}$ we refrain
from listing them here. The first 200 terms of the threshold expansion of all
form factors are given in the ancillary files. After this we can find
recursions obeyed by the sequences of rational numbers associated to the each
combination of a given constant (including the known constants and the new
ones, $\tilde{\kappa}_{i}$) and a power of the log. For example, the sequence
of rational number associated to the term $\pi\ln^{2}(z)$ in the non-solvable
parts (at the expansion point $x=1$) of $F_{V,1}^{(0)}(z)$ are given by

$\displaystyle\biggl{\\{}0,0,0,\frac{27008}{27},0,\frac{2752}{81},0,-\frac{339913}{1215},0,-\frac{27836749}{170100},0,-\frac{2883493363}{76204800},0,$
$\displaystyle\phantom{\biggl{\\{}}-\frac{33749896783}{3772137600},0,-\frac{32996303383481}{15343379251200},0,-\frac{749728845867049}{1436140297912320},0,\ldots\biggr{\\}}.$
(4.42)

In other words, the $n$th term in the sequence above is the rational number
multiplying $x^{-7+n}\pi\ln^{2}(z)$ in the expansion, starting from $n=1$. As
usual, the first few terms must be dropped in order to find a recursion.
Notice that all terms with $n$ even are equal to zero. This is a consequence
of the patterns described above and requires us to find a sequence by
considering the terms with $n$ odd only. We did so, and found a recurrence of
order five,

$\sum_{i=0}^{5}\tilde{p}_{i}(n)f_{3}(n+i)=0,$ (4.43)

where $f_{3}(n)$ is the $n$th term of the sequence given above after dropping
the first seven terms and all the zeroes, and the $\tilde{p}_{i}(n)$’s are
polynomials in $n$. We found the following solution

$\displaystyle f_{3}(n)=$ (4.44)
$\displaystyle-\frac{2^{1-4n}T_{3}}{27(1+n)(2+n)(3+n)(1+2n)^{2}(3+2n)^{2}(5+2n)}\binom{2n}{n}-\frac{\big{(}349+1164n+612n^{2}\big{)}2^{5-2n}}{27(1+2n)(3+2n)}$
$\displaystyle-\frac{\big{(}41967+66774n+32884n^{2}+5768n^{3}\big{)}2^{1-4n}\binom{2n}{n}}{27(1+n)(2+n)(3+n)}S_{\\{2,1,2\\}}(n)-\frac{1}{27}2^{8-2n}\sum_{i_{1}=1}^{n}\frac{2^{-2i_{1}}\binom{2i_{1}}{i_{1}}}{\big{(}1+2i_{1}\big{)}^{2}}$
$\displaystyle+\Biggl{[}-\frac{5}{27}2^{6-2n}-\frac{3\big{(}-7+4n+4n^{2}\big{)}2^{5-4n}\binom{2n}{n}}{(1+2n)(3+2n)}-\frac{5}{27}2^{6-2n}\sum_{i_{1}=1}^{n}\frac{2^{-2i_{1}}\binom{2i_{1}}{i_{1}}}{1+2i_{1}}\Biggr{]}S_{\\{2,1,1\\}}(n)$
$\displaystyle-\frac{5\big{(}65+220n+116n^{2}\big{)}2^{5-2n}\sum_{i_{1}=1}^{n}\frac{2^{-2i_{1}}\binom{2i_{1}}{i_{1}}}{1+2i_{1}}}{27(1+2n)(3+2n)}-\frac{71}{27}2^{5-2n}\sum_{i_{1}=1}^{n}\frac{2^{-2i_{1}}\binom{2i_{1}}{i_{1}}S_{\\{2,1,1\\}}(i_{1})}{1+2i_{1}},$

with

$\displaystyle T_{3}$ $\displaystyle=$ $\displaystyle
434624n^{8}+5219648n^{7}+26709296n^{6}+75033264n^{5}+124288324n^{4}+121056044n^{3}$
(4.45) $\displaystyle+64493993n^{2}+15127189n+517179\,.$

Here $S_{\\{2,1,1\\}}(n)$ and $S_{\\{2,1,2\\}}(n)$ are cyclotomic harmonic
sums [37], which are defined as

$S_{\\{a,b,c\\}}(n)=\sum_{k=1}^{n}\frac{({\rm sign}(c))^{k}}{(ak+b)^{|c|}}.$
(4.46)

Resumming these expressions with the help of the HarmonicSums command Compute
GeneratingFunction one obtains expressions depending on iterative integrals of
the kind

$G\left(\\{f_{1}(\tau),f_{2}(\tau),\ldots,f_{n}(\tau)\\},z\right)=\int_{0}^{z}d\tau_{1}\,\,f_{1}(\tau_{1})G\left(\\{f_{2}(\tau),\ldots,f_{n}(\tau)\\},\tau_{1}\right),$
(4.47)

with

$\displaystyle F_{3}(x)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}f_{3}(i)x^{i}=-\frac{64(1266+43x)}{81x^{2}}-\frac{16\text{G}\big{(}\sqrt{4-\tau}\sqrt{\tau};x\big{)}R_{6}}{27x^{5/2}(4-x)}+\frac{4\sqrt{4-x}R_{7}}{27(4-x)x^{2}}$
(4.48)
$\displaystyle-48(-2+x)\sqrt{4-x}\text{G}\left(\frac{1}{4-\tau};x\right)-\frac{320(2-x)^{2}}{27x^{3/2}(4-x)}\text{G}\left(\frac{1}{4-\tau},\sqrt{4-\tau}\sqrt{\tau};x\right)$
$\displaystyle-\frac{2848(2-x)^{2}}{27x^{3/2}(4-x)}\text{G}\left(\frac{1}{\tau},\sqrt{4-\tau}\sqrt{\tau};x\right)-\frac{96(2-x)^{2}}{x^{3/2}(4-x)}\text{G}\left(\sqrt{4-\tau}\sqrt{\tau},\frac{1}{4-\tau};x\right)$
$\displaystyle-\frac{32\big{(}4836-1636x+19x^{2}+67x^{3}\big{)}\sqrt{4-x}}{27(4-x)x^{3}}\text{G}\left(\sqrt{4-\tau}\sqrt{\tau},\sqrt{4-\tau}\sqrt{\tau};x\right),$

and

$\displaystyle R_{6}$ $\displaystyle=$ $\displaystyle
67x^{5}-383x^{4}-1202x^{3}+13712x^{2}-26888x+1184,$ (4.49) $\displaystyle
R_{7}$ $\displaystyle=$ $\displaystyle
67x^{6}-517x^{5}-104x^{4}+12936x^{3}-39828x^{2}+1928x+13504.$ (4.50)

Note that here the sum starts at $i=0$, unlike the case for $f_{1(2)}(i)$ in
Secton 3. The corresponding expression is

$\displaystyle
F_{\pi\ln^{2}(z),V,1}^{(0)}=\frac{27008}{27z^{3}}+\frac{2752}{81z}+zF_{3}(z^{2})$
(4.51)

The $G$–functions are given as polynomials of rational terms and harmonic
polylogarithms in the variable

$\displaystyle
u=\frac{1}{2}\left(2+iz\sqrt{4-z^{2}}-z^{2}\right)=-\frac{1}{x}.$ (4.52)

One obtains

$\displaystyle G\left(\left\\{\sqrt{4-\tau}\sqrt{\tau}\right\\},z^{2}\right)$
$\displaystyle=$ $\displaystyle i\left[-\frac{1}{2u^{2}}+\frac{u^{2}}{2}-2{\rm
H}_{0}(u)\right],$ (4.53) $\displaystyle
G\left(\left\\{\frac{1}{\tau},\sqrt{4-\tau}\sqrt{\tau}\right\\},z^{2}\right),$
$\displaystyle=$ $\displaystyle
i\Biggl{[}-\frac{(1-u^{2})\big{(}1+4u+u^{2}\big{)}}{4u^{2}}+4{\rm
H}_{0}(u){\rm H}_{1}(u)-4{\rm H}_{0,1}(u)$ (4.54)
$\displaystyle+4\zeta_{2}+{\rm H}_{0}(u)+{\rm H}_{0}^{2}(u)\Biggr{]},$
$\displaystyle
G\left(\left\\{\frac{1}{4-\tau},\sqrt{4-\tau}\sqrt{\tau}\right\\},z^{2}\right)$
$\displaystyle=$ $\displaystyle
i\Biggl{[}\frac{(1-u^{2})\big{(}1-4u+u^{2}\big{)}}{4u^{2}}-{\rm
H}_{0}^{2}(u)-4{\rm H}_{0,-1}(u)$ (4.55) $\displaystyle+2\zeta_{2}+{\rm
H}_{0}(u)\big{(}-1+4{\rm H}_{-1}(u)\big{)}\Biggr{]}$ $\displaystyle
G\left(\left\\{\sqrt{4-\tau}\sqrt{\tau},\frac{1}{4-\tau}\right\\},z^{2}\right)$
$\displaystyle=$ $\displaystyle
G\left(\left\\{\frac{1}{4-\tau}\right\\},z^{2}\right)G\left(\left\\{\sqrt{4-\tau}\sqrt{\tau}\right\\},z^{2}\right)$
(4.56)
$\displaystyle-G\left(\left\\{\frac{1}{4-\tau},\sqrt{4-\tau}\sqrt{\tau}\right\\},z^{2}\right),$

by shuffling [41], where

$\displaystyle G\left(\left\\{\frac{1}{4-\tau}\right\\},z^{2}\right)$
$\displaystyle=$ $\displaystyle-\ln\left[1-\frac{z^{2}}{4}\right],$ (4.57)

Iterative $G$-functions over root–valued alphabets were studied in [38, 75].

After adding up all the terms for which the recurrences are solvable at the
expansion point $\hat{s}=4$, we find that the previously non-solvable parts at
the expansions point $x=1$ in Eq. (2.25) can be resummed as follows,

$\displaystyle
F^{(0)}_{V,1,1}(z)+\zeta_{2}F^{(0)}_{V,1,2}(z)+\zeta_{3}F^{(0)}_{V,1,3}(z)=$
(4.58)
$\displaystyle-\frac{32\ln(2)\pi^{3}\frac{1}{\sqrt{4-z^{2}}}\ln\big{(}1-\frac{z^{2}}{4}\big{)}Q_{13}}{27z^{5}}+\frac{64\ln(2)\pi^{3}\frac{1}{\sqrt{4-z^{2}}}\ln(z)Q_{24}}{27z^{5}}-\frac{4\ln^{2}(2)\pi\sqrt{4-z^{2}}Q_{42}}{27z}$
$\displaystyle-\frac{8\ln(2)\pi\sqrt{4-z^{2}}\ln(z)Q_{42}}{9z}-\frac{4\pi^{3}\frac{1}{\sqrt{4-z^{2}}}\ln(z)Q_{48}}{27z^{3}}-\frac{32\pi\frac{1}{\sqrt{4-z^{2}}}\ln^{3}(z)Q_{51}}{81z^{3}}$
$\displaystyle-\frac{64\pi^{2}\ln^{2}(z)Q_{55}}{27(z-2)z^{4}(z+2)}-\frac{16\pi\frac{\zeta_{3}}{\sqrt{4-z^{2}}}Q_{64}}{729z^{5}}-\frac{2\ln(2)\pi^{3}\frac{1}{\sqrt{4-z^{2}}}Q_{65}}{243z^{5}}+\frac{4\pi\frac{1}{\sqrt{4-z^{2}}}\ln^{2}(z)Q_{73}}{27z^{3}}$
$\displaystyle-\frac{\pi^{2}\ln(z)Q_{80}}{486(z-2)z^{2}(z+2)}+\ln^{2}(2)\pi^{3}\Biggl{[}-\frac{32\big{(}5981-3570z^{2}+300z^{4}\big{)}\frac{1}{\sqrt{4-z^{2}}}}{243z^{3}}$
$\displaystyle-\frac{64\big{(}-12+5z^{2}\big{)}\frac{1}{\sqrt{4-z^{2}}}\ln\big{(}1-\frac{z^{2}}{4}\big{)}}{3z^{5}}\Biggr{]}+\ln^{4}(2)\pi\Biggl{[}-\frac{128\big{(}85+24z^{2}+6z^{4}\big{)}\frac{1}{\sqrt{4-z^{2}}}}{243z^{3}}$
$\displaystyle+\frac{64\big{(}-12+5z^{2}\big{)}\frac{1}{\sqrt{4-z^{2}}}\ln\big{(}1-\frac{z^{2}}{4}\big{)}}{3z^{5}}\Biggr{]}+{\rm
Li}_{4}\left(\frac{1}{2}\right)\pi\Biggl{[}-\frac{1024\big{(}85+24z^{2}+6z^{4}\big{)}\frac{1}{\sqrt{4-z^{2}}}}{81z^{3}}$
$\displaystyle+\frac{512\big{(}-12+5z^{2}\big{)}\frac{1}{\sqrt{4-z^{2}}}\ln\big{(}1-\frac{z^{2}}{4}\big{)}}{z^{5}}\Biggr{]}+\pi^{5}\Biggl{[}-\frac{4\big{(}20915-14010z^{2}+4836z^{4}\big{)}\frac{1}{\sqrt{4-z^{2}}}}{3645z^{3}}$
$\displaystyle-\frac{8\big{(}12+43z^{2}\big{)}\frac{1}{\sqrt{4-z^{2}}}\ln\big{(}1-\frac{z^{2}}{4}\big{)}}{45z^{5}}\Biggr{]}+\Biggl{[}-\frac{64\ln^{3}(2)\pi\big{(}z^{2}-2\big{)}^{2}}{9(z-2)z^{2}(z+2)}+\frac{128\pi^{2}\frac{1}{\sqrt{4-z^{2}}}\ln^{2}(z)Q_{8}}{27z^{5}}$
$\displaystyle-\frac{224\pi\big{(}z^{2}-2\big{)}\zeta_{3}Q_{16}}{27(z-2)z^{6}(z+2)}-\frac{8\pi^{3}\big{(}z^{2}-2\big{)}\ln(z)Q_{29}}{27(z-2)z^{6}(z+2)}-\frac{16\ln(2)\pi^{3}Q_{52}}{27(z-2)z^{6}(z+2)}$
$\displaystyle+\frac{16\ln^{2}(2)\pi
Q_{56}}{27(z-2)z^{4}(z+2)}+\frac{32\ln(2)\pi\ln(z)Q_{56}}{9(z-2)z^{4}(z+2)}+\frac{16\pi\ln^{2}(z)Q_{57}}{27(z-2)z^{4}(z+2)}+\frac{\pi^{2}\frac{1}{\sqrt{4-z^{2}}}\ln(z)Q_{62}}{81z^{5}}$
$\displaystyle-\frac{64\ln^{2}(2)\pi\big{(}z^{2}-2\big{)}^{2}\ln(z)}{(z-2)z^{2}(z+2)}+\frac{64\ln(2)\pi^{2}\big{(}-30+8z^{2}-3z^{4}+z^{6}\big{)}\frac{1}{\sqrt{4-z^{2}}}\ln(z)}{9z^{5}}$
$\displaystyle-\frac{192\ln(2)\pi\big{(}z^{2}-2\big{)}^{2}\ln^{2}(z)}{(z-2)z^{2}(z+2)}-\frac{10432\pi\big{(}z^{2}-2\big{)}^{2}\ln^{3}(z)}{81(z-2)z^{2}(z+2)}\Biggr{]}G\left(\sqrt{4-\tau}\sqrt{\tau};z^{2}\right)$
$\displaystyle+\Biggl{[}-\frac{16\ln^{2}(2)\pi\frac{1}{\sqrt{4-z^{2}}}Q_{11}}{27z^{5}}-\frac{32\ln(2)\pi\frac{1}{\sqrt{4-z^{2}}}\ln(z)Q_{11}}{9z^{5}}-\frac{16\pi\frac{1}{\sqrt{4-z^{2}}}\ln^{2}(z)Q_{22}}{27z^{5}}$
$\displaystyle+\frac{2\pi^{2}\ln(z)Q_{43}}{9(z-2)z^{6}(z+2)}+\frac{8\ln(2)\pi^{3}\big{(}-36+7z^{2}\big{)}\frac{1}{\sqrt{4-z^{2}}}}{z^{5}}\Biggr{]}G\big{(}\sqrt{4-\tau}\sqrt{\tau};z^{2}\big{)}^{2}$
$\displaystyle+\Biggl{[}\frac{64\ln^{2}(2)\pi\big{(}z^{2}-2\big{)}^{2}}{27(z-2)z^{2}(z+2)}+\frac{128\ln(2)\pi\big{(}z^{2}-2\big{)}^{2}\ln(z)}{9(z-2)z^{2}(z+2)}+\frac{320\pi\big{(}z^{2}-2\big{)}^{2}\ln^{2}(z)}{27(z-2)z^{2}(z+2)}$
$\displaystyle+\frac{64\pi^{2}\big{(}3+z^{2}\big{)}\big{(}6-4z^{2}+z^{4}\big{)}\frac{1}{\sqrt{4-z^{2}}}\ln(z)}{27z^{5}}\Biggr{]}G\left(\frac{1}{4-\tau},\sqrt{4-\tau}\sqrt{\tau};z^{2}\right)$
$\displaystyle+\Biggl{[}\frac{32\ln^{2}(2)\pi\big{(}z^{2}-2\big{)}^{2}}{3(z-2)z^{2}(z+2)}-\frac{16\pi^{2}\frac{1}{\sqrt{4-z^{2}}}\ln(z)Q_{18}}{9z^{5}}+\frac{64\ln(2)\pi\big{(}z^{2}-2\big{)}^{2}\ln(z)}{(z-2)z^{2}(z+2)}$
$\displaystyle+\frac{2848\pi\big{(}z^{2}-2\big{)}^{2}\ln^{2}(z)}{27(z-2)z^{2}(z+2)}\Biggr{]}G\left(\frac{1}{\tau},\sqrt{4-\tau}\sqrt{\tau};z^{2}\right)+\Biggl{[}\frac{32\ln^{2}(2)\pi\big{(}z^{2}-2\big{)}^{2}}{3(z-2)z^{2}(z+2)}$
$\displaystyle+\frac{64\ln(2)\pi\big{(}z^{2}-2\big{)}^{2}\ln(z)}{(z-2)z^{2}(z+2)}+\frac{96\pi\big{(}z^{2}-2\big{)}^{2}\ln^{2}(z)}{(z-2)z^{2}(z+2)}\Biggr{]}G\left(\sqrt{4-\tau}\sqrt{\tau},\frac{1}{4-\tau};z^{2}\right)$
$\displaystyle+\frac{4096{\rm
Li}_{4}\left(\frac{1}{2}\right)\ln(2)\pi\frac{1}{\sqrt{4-z^{2}}}}{3z^{3}}+\frac{512\ln(2)^{5}\pi\frac{1}{\sqrt{4-z^{2}}}}{9z^{3}}-\frac{512\ln^{3}(2)\pi^{3}\frac{1}{\sqrt{4-z^{2}}}}{9z^{3}}-\frac{1600\ln(2)\pi^{5}\frac{1}{\sqrt{4-z^{2}}}}{27z^{3}}$
$\displaystyle-\frac{\pi^{2}\big{(}-36+11z^{2}\big{)}\kappa_{2}\frac{1}{\sqrt{4-z^{2}}}}{36z^{5}}-\frac{\big{(}-36+11z^{2}\big{)}\kappa_{9}\frac{1}{\sqrt{4-z^{2}}}}{6z^{5}}+\frac{32}{9}\ln^{3}(2)\pi
z\big{(}z^{2}-2\big{)}\sqrt{4-z^{2}}$
$\displaystyle+\frac{248}{27}\ln^{2}(2)\pi
z\big{(}4-z^{2}\big{)}^{3/2}+\frac{2166080\big{(}-36+11z^{2}\big{)}\frac{\zeta_{3}}{\sqrt{4-z^{2}}}}{729\pi
z^{5}}$ $\displaystyle-\frac{1792{\rm
Li}_{4}\left(\frac{1}{2}\right)\big{(}-36+11z^{2}\big{)}\frac{\zeta_{3}}{\sqrt{4-z^{2}}}}{\pi
z^{5}}-\frac{2034368\ln(2)\big{(}-36+11z^{2}\big{)}\frac{\zeta_{3}}{\sqrt{4-z^{2}}}}{729\pi
z^{5}}$
$\displaystyle-\frac{17024\ln^{2}(2)\big{(}-36+11z^{2}\big{)}\frac{\zeta_{3}}{\sqrt{4-z^{2}}}}{243\pi
z^{5}}+\frac{7168\ln^{3}(2)\big{(}-36+11z^{2}\big{)}\frac{\zeta_{3}}{\sqrt{4-z^{2}}}}{27\pi
z^{5}}$
$\displaystyle-\frac{224\ln^{4}(2)\big{(}-36+11z^{2}\big{)}\frac{\zeta_{3}}{\sqrt{4-z^{2}}}}{3\pi
z^{5}}$
$\displaystyle+\frac{64256\ln(2)\pi\big{(}-36+11z^{2}\big{)}\frac{\zeta_{3}}{\sqrt{4-z^{2}}}}{81z^{5}}-\frac{736\ln^{2}(2)\pi\big{(}-36+11z^{2}\big{)}\frac{\zeta_{3}}{\sqrt{4-z^{2}}}}{3z^{5}}$
$\displaystyle+\frac{\pi^{3}\big{(}-64836+25451z^{2}\big{)}\frac{\zeta_{3}}{\sqrt{4-z^{2}}}}{135z^{5}}+\frac{17\big{(}-36+11z^{2}\big{)}\kappa_{1}\frac{\zeta_{3}}{\sqrt{4-z^{2}}}}{72z^{5}}-\frac{\big{(}-36+11z^{2}\big{)}\kappa_{3}\frac{\zeta_{3}}{\sqrt{4-z^{2}}}}{9z^{5}}$
$\displaystyle-\frac{49112\big{(}-36+11z^{2}\big{)}\frac{\zeta_{3}^{2}}{\sqrt{4-z^{2}}}}{81\pi
z^{5}}-\frac{640\pi^{4}\big{(}z^{2}-2\big{)}^{2}\ln(z)}{27(z-2)z^{2}(z+2)}-\frac{360\pi\frac{\zeta_{5}}{\sqrt{4-z^{2}}}}{z^{3}}$
$\displaystyle-\frac{32\ln(2)\pi^{2}\big{(}-16+10z^{2}-5z^{4}+z^{6}\big{)}\ln(z)}{9z^{2}}+32\ln^{2}(2)\pi
z\big{(}z^{2}-2\big{)}\sqrt{4-z^{2}}\ln(z)$
$\displaystyle+\frac{496}{9}\ln(2)\pi
z\big{(}4-z^{2}\big{)}^{3/2}\ln(z)+\frac{4\pi^{2}\big{(}-12+5z^{2}\big{)}\frac{1}{\sqrt{4-z^{2}}}G\big{(}\sqrt{4-\tau}\sqrt{\tau};z^{2}\big{)}^{3}\ln(z)}{3z^{5}}$
$\displaystyle+96\ln(2)\pi
z\big{(}z^{2}-2\big{)}\sqrt{4-z^{2}}\ln^{2}(z)+\frac{488}{9}\pi
z\big{(}4-z^{2}\big{)}^{3/2}\ln^{2}(z)-\frac{4096\pi^{2}\big{(}z^{2}-2\big{)}^{2}\ln^{3}(z)}{81(z-2)z^{2}(z+2)}$
$\displaystyle+\frac{16}{3}\ln^{2}(2)\pi
z\big{(}z^{2}-2\big{)}\sqrt{4-z^{2}}\ln\big{(}1-\frac{z^{2}}{4}\big{)}+32\ln(2)\pi
z\big{(}z^{2}-2\big{)}\sqrt{4-z^{2}}\ln(z)\ln\big{(}1-\frac{z^{2}}{4}\big{)}$
$\displaystyle+48\pi
z\big{(}z^{2}-2\big{)}\sqrt{4-z^{2}}\ln^{2}(z)\ln\big{(}1-\frac{z^{2}}{4}\big{)}+\tilde{F}^{(0),\rm
rest}_{V,1}(z)\,.$

The polynomials $Q_{i}$ are given in Appendix D, where also the resummed
expressions in $z$ for the remaining form factors are presented.

###  4.3 The four-particle threshold

The differential equations in the case of the four-particle threshold were
derived from the ones guessed using the expansions at $\hat{s}=0$ by doing the
change of variables $\tilde{z}$, after which we matched with the expansions at
$\hat{s}\rightarrow+\infty$ ($x=0$). In all cases the matching was done at a
set of points given by Eq. (4.12) using $\delta=1/2000$. No derivatives were
used. Since the expansions at $\hat{s}=16$ converge in the region
$4<\hat{s}<28$, and the high energy expansion converges for $\hat{s}>16$, we
had to choose a value of $\hat{s}_{0}$ in Eq. (4.12) in the region
$16<\hat{s}_{0}<28$. We used $\hat{s}_{0}=20$ as the starting point.

We found agreement with two of the statements made in Ref. [16, 17], namely,
that these expansions contain no logarithms in the variable $\tilde{z}$, and
that the first non-analytic terms (in other words, odd powers of $\tilde{z}$)
start at order 7 in the case of the axial-vector and scalar form factors, and
at order 9 in the case of the vector and pseudo-scalar form factors. The
authors of [16, 17] did find powers of logarithms as high as
$\ln^{5}(\tilde{z})$, together with powers of $\tilde{z}$ as low as
$\tilde{z}^{3}$ at the level of the master integrals, which then disappear
from the final results. Since we do not deal with master integrals in our
method, at no point in our calculations do we find such terms, which makes
matching with the high energy expansion easier in our case. We can even
pinpoint from which of the form factors $F_{I,k}^{(0)}(\tilde{z})$ do the
first non-analytic terms come from. The non-solvable parts
$F_{I,2}^{(0)}(\tilde{z})$ turn out to be analytic for all values of $I$. In
all other cases, the first non-analytic terms start at $\tilde{z}^{9}$, with
the exception of $F_{a,1,1}^{(0)}(\tilde{z})$, $F_{a,1,3}^{(0)}(\tilde{z})$,
$F_{s,1}^{(0)}(\tilde{z})$ and $F_{s,3}^{(0)}(\tilde{z})$, where they start at
order $\tilde{z}^{7}$. As examples, the non-solvable (at $x=1$) parts of the
vector form factor, $F_{v,1,1}^{(0)}(\tilde{z})$ and
$F_{v,1,3}^{(0)}(\tilde{z})$ have the following numerical expansions at
$\tilde{z}=0$,

$\displaystyle F_{v,1,1}^{(0)}(\tilde{z})$ $\displaystyle=$ $\displaystyle
708902.27306400652676-1166.85387707523430i$ (4.59)
$\displaystyle-(61456.423805594065279-15405.248917971629065i)\tilde{z}^{2}$
$\displaystyle+(1716.6533220464368589-30.2703684431181752i)\tilde{z}^{4}$
$\displaystyle+(11.2780421205283601446+4.3654180757157056768i)\tilde{z}^{6}$
$\displaystyle+(0.74166620892322374519+0.87946094505024891071i)\tilde{z}^{8}$
$\displaystyle+(0.00029237832702742120027-0.00015868965691879171216i)\tilde{z}^{9}$
$\displaystyle+(0.063605792722744201987+0.086239707160403484962i)\tilde{z}^{10}$
$\displaystyle+(0.000083577138055319044650-0.000045361868983569336325i)\tilde{z}^{11}$
$\displaystyle+(0.0055226467252660130699+0.0075199442142108863165i)\tilde{z}^{12}$
$\displaystyle+(0.000016701125786592152314-9.0646114169171\times
10^{-6}i)\tilde{z}^{13}$
$\displaystyle+(0.00047286013902545145990+0.00063297686865989569848i)\tilde{z}^{14}$
$\displaystyle+O\left(\tilde{z}^{15}\right)\,,$

and

$\displaystyle F_{v,1,3}^{(0)}(\tilde{z})$ $\displaystyle=$
$\displaystyle-(525309.50058104203958+59965.62896057212016i)$ (4.60)
$\displaystyle+(38742.883772468588201+5783.453693303291125i)\tilde{z}^{2}$
$\displaystyle-(960.03909739184249814+38.76539648430706459i)\tilde{z}^{4}$
$\displaystyle-(7.5313243403421680570+0.2607012992726984795i)\tilde{z}^{6}$
$\displaystyle-(0.66891050212132794057-0.02251916817914504430i)\tilde{z}^{8}$
$\displaystyle+(5.98\times
10^{-192}+0.000132015095543045899654i)\tilde{z}^{9}$
$\displaystyle-(0.056139396989125489861-0.002911370516926165654i)\tilde{z}^{10}$
$\displaystyle+(1.71\times
10^{-192}+0.000037736873241471453874i)\tilde{z}^{11}$
$\displaystyle-(0.0046441851980261141902-0.0002618233295226892918i)\tilde{z}^{12}$
$\displaystyle+(3.42\times 10^{-193}+7.5409170673126\times
10^{-6}i)\tilde{z}^{13}$
$\displaystyle+(-0.00038214879532394872786+0.00002100556849783558601i)\tilde{z}^{14}$
$\displaystyle+O\left(\tilde{z}^{15}\right)\,.$

Notice that in the case of $F_{v,1,3}^{(0)}(\tilde{z})$ the real parts of the
coefficients of the odd powers in $\tilde{z}$ are very small, namely, of the
order of $10^{-192}$ or lower. Clearly, the real parts of these coefficients
are actually equal to zero, and the values obtained indicate that we are
computing the coefficients with a precision of around 200 digits. To get the
final form factors, we also need to combine these results using Eq. (2.25). It
can be shown that the imaginary parts of the coefficients of the odd powers in
$\tilde{z}$ of the combination
$F_{v,1,1}^{(0)}(\tilde{z})+\zeta_{3}F_{v,1,3}^{(0)}(\tilde{z})$ are of the
order of $10^{-166}$ or lower. Again, this indicates that these imaginary
parts actually cancel in the combination, and it is also a strong indication
that the calculation is correct, since such cancellations are unlikely to
happen otherwise.

At this point, we could proceed in a similar way as we did with previous
expansions and use this type of information about the coefficients to improve
the precision by matching at better points (that is, choosing better values of
$\hat{s}_{0}$ and $\delta$). The main reason for doing this would be to try to
determine the base coefficients in terms of known constants, as we have done
in the case of the expansions at the other points. However, this turns out to
be impractical in this case. Even the simplest coefficients of solvable parts
are difficult to determine using the PSLQ algorithm, due to the large number
of constants that would need to be included in a search. These constants
involve all sorts of harmonic polylogarithms of weight up to 5 evaluated at

$\displaystyle\xi=7-4\sqrt{3},$ (4.61)

on top of the constants that appeared in previous expansions, such as
$\zeta_{k}$, $a_{4}$, $l_{2}^{k}$ and their products. For example, the
coefficient of $\tilde{z}^{0}$ of the term proportional to $n_{h}^{2}$ in the
vector form factor $F_{v,1}^{(0)}(\tilde{z})$ (which is much simpler than any
of the terms proportional to $n_{h}$) contains the following constants and
their products,

$\displaystyle\biggl{\\{}{\rm
H}_{0,1}(\xi),l_{2},\,\pi,\,\zeta_{2},\,\zeta_{3},\,\frac{1}{\sqrt{3}},\,{\rm
H}_{0}(\xi),\,{\rm H}_{1}(\xi),\,{\rm H}_{-1}(\xi),\,{\rm H}_{0,1}(\xi),\,{\rm
H}_{0,-1}(\xi),\,{\rm H}_{0,0,1}(\xi),\,$
$\displaystyle\phantom{\biggl{\\{}}{\rm H}_{0,0,-1}(\xi),\,{\rm
H}_{0,1,1}(\xi),\,{\rm H}_{0,0,0,1}(\xi),\,{\rm H}_{0,0,0,-1}(\xi),\,{\rm
H}_{0,0,1,1}(\xi),\,{\rm H}_{0,1,1,1}(\xi)\biggr{\\}}.$ (4.62)

The non-solvable parts of the form factors will most likely be much more
complicated, containing these and new constants, and we would need to
determine the coefficients numerically with an unattainable precision in order
to be able to do a PSLQ search. Fortunately, the expansions at $\hat{s}=16$,
which are very important since they are the only ones that cover the region
$8<\hat{s}<16$, are also very well behaved and rather smooth at the expansion
point, and the numerical results we have obtained are enough for most
applications.

With all the expansion thus far calculated, we cover the entire range of
$\hat{s}$. In an ancillary file to this paper we provide the representations
for the expansion around the four singular points for phenomenological use.

##  5 Numerical Results

In the following we illustrate non–solvable constant parts, cf. Ref. [15], of
the real part to the different form factors
${F_{V,1},F_{V,2},F_{A,1},F_{A,2},F_{S}}$ and ${F_{P}}$ as a function of
$\hat{s}=q^{2}/m^{2}$ for the constant

Figure 3: Real part of the non-solvable contribution for the vector form
factor $F_{V,1}$ in dependence on $\hat{s}$ and $x$. The range of $x\in[0,1]$
corresponds to $\hat{s}\in[-\infty,0]$. The different line styles correspond
to the various expansions. The vertical dashed lines indicate the
(pseudo-)thresholds. Left panel: the overlapping expansions are performed
around $\hat{s}=-\infty,0,4,16$ and $\hat{s}=\infty$. Right panel: full line:
expansion around $x=0$; dashed line: expansion around $x=1$.

contribution at $O(a_{s}^{3})$ numerically. Here we use the first 200
expansion terms around $x=1$, $x=0$, and the (pseudo) thresholds $\hat{s}=16$
and $4$. For this approach we estimated our numerical accuracy to be better
than $10^{-40}$. The numerical work is done by using Mathematica. For
harmonic, generalized harmonic [76, 77] and cyclotomic harmonic polylogarithms
[77, 78] there are also different other numerical implementations.

Figure 4: The same as Figure 3 for ${F_{V,2}}$.

Figure 5: The same as Figure 3 for ${F_{A,1}}$.

Figure 6: The same as Figure 3 for ${F_{A,2}}$.

In Figures 3–8 we use different rescalings in $\hat{s}$ in part to allow for
better visibility. Figures 3, 4 show the results for the vector form factors,
Figures 5, 6 for the axial–vector form factors, and Figures 7 and 8 for the
scalar and pseudo–scalar form factor, respectively. The left panels show the
contributions as a function of $\hat{s}$, while in the right panel we
illustrate the form factors in the region $x\in[0,1]$. Similar numerical
results are obtained for the corresponding imaginary parts in the non–solvable
case, which we are not illustrating.

Figure 7: The same as Figure 3 for ${F_{S}}$.

Figure 8: The same as Figure 3 for ${F_{P}}$.

The singularities of the form factors in the variable $\hat{s}$ are indicated.
In the above figures it is shown, that the series expansions used do
significantly overlap, which allows for a continuous representation.
Concerning the expansion around $x=0$, the singularity left to this value at
$x=4\cdot\sqrt{3}-7\approx-0.0718$ requires that the expansion around $x=1$
has to be used even below $7-4\cdot\sqrt{3}$. We can extend our series
representations to higher orders, to meet even higher accuracy.

In Figures 9–14 we illustrate the real and imaginary parts of the constant
$O(a_{s}^{3})$ contributions to the vector (Figures 9, 10), axial–vector
(Figures 11, 12), scalar (Figure 13), and pseudo–scalar form factor (Figure
14) as functions of $\hat{s}$.

Figure 9: Real and imaginary part of the complete heavy-fermionic
contributions to the vector form factor $F_{V,1}$ in dependence on $\hat{s}$.

Figure 10: Real and imaginary part of the complete heavy-fermonic
contributions to the vector form factor $F_{V,2}$ in dependence on $\hat{s}$.

Figure 11: Real and imaginary part of the complete heavy-fermonic
contributions to the axial-vector form factor $F_{A,1}$ in dependence on
$\hat{s}$.

Figure 12: Real and imaginary part of the complete heavy-fermonic
contributions to the axial-vector form factor $F_{A,2}$ in dependence on
$\hat{s}$.

Figure 13: Real and imaginary part of the complete heavy-fermonic
contributions to the scalar form factor $F_{S}$ in dependence on $\hat{s}$.

Figure 14: Real and imaginary part of the complete heavy-fermonic
contributions to the pseudo-scalar form factor $F_{P}$ in dependence on
$\hat{s}$.

Finally, we would like to discuss the numerical comparison of the present
results with those of Ref. [16]. We compared the results at
$\hat{s}=\\{-300,-20,-10,-5,-0.1,-0.001,$
$-0.0001,0.00001,0.1,3.9,3.9999,10,15.9999,16.0001,16.1,20,39.9999,40\\}$
using their Mathematica files. In general we found a relative agreement of
$O(10^{-10})$ or better, except for $\hat{s}=39.9999$, where the deviation was
in the range $[6.2\cdot 10^{-6},1.6\cdot 10^{-9}]$.

Furthermore, we compared the representations of [16], Eq. (35–40), in the high
energy limit $s\rightarrow-\infty$ to our results for $n_{h}=1$. Setting the
color factors to those of QCD we obtain the following relative deviations

$\displaystyle\left|\frac{{F_{I}^{\mathrm{KIT}}}-{F_{I}}}{{F_{I}}}\right|\in[2.27\cdot
10^{-7},1.13\cdot 10^{-4}],$ (5.1)

for $I=\\{{(V,1),(V,2),(A,1),(A,2),S,P}\\}$.

We provide our numerical representation in Mathematica format at [79].

##  6 Conclusions

We have calculated analytic representations for the quarkonic contributions to
the massive three–loop form factors for different currents based on
non–trivial computer algebra methods, including guessing algorithms to derive
recursion relations and differential equation and to determine representations
in terms of known constants. Up to two–loop order, the massive form factors
can be represented in terms of harmonic polylogarithms. From three–loop order
onward there are two principal contributions [15]. One part results from
first–order factorizable recurrences which in $x$–space can be represented in
terms of harmonic and cyclotomic harmonic polylogarithms. The second part is
related to the remaining non–first order factorizing recurrences. We first
consider their representation in terms of a series expansion around $x=1$,
which we then match with the symbolic series expansion around $x=0$. The
latter also contains a finite number of series–modulated logarithmic
contributions. This series representation has a convergence radius of
$x=7-4\sqrt{3}$, due to the pseudo–threshold at $q^{2}/m^{2}=16$. Therefore,
the matching of both series has to be performed for small positive values of
$x$. We use a rather large number of expansion coefficients for this, at very
high numerical precision. Thanks to this, the unknown expansion coefficients
at $x=0$ can be determined by using the PSLQ algorithm. In this way, we
determine the series representations for all form factors around $x=0$. The
coefficients turn out to depend only on multiple zeta values linearly combined
over $\mathbb{Q}$. Moreover, the $x$–series can be resummed for a series of
constants into harmonic polylogarithms in $x$ using the packages Sigma and
HarmonicSums. This applies to cases in which the corresponding recurrence
(guessed with the package ore_algebra) factorizes to first order. We could
determine all recurrences, which allows an even faster evaluation of the
corresponding series, whenever needed.

This procedure was repeated for the other two singularities (thresholds)
around $q^{2}/m^{2}=4$ and $q^{2}/m^{2}=16$. Here, however, unlike in the high
energy limit $x=0$, our analysis showed that multiple zeta values are not
sufficient for the corresponding series representations, which was expected
due to elliptic (or even higher order) master integrals. We determined the new
constants $\tilde{\kappa}_{i}$ at high numerical precision for further special
analyses. Our series expansions can be used in phenomenological and
experimental applications. Still we provide analytic results in terms of
series representations and in some cases also resumming the respective series
into special functions. Our representations are given in terms of expansions
around the four singular points to very high precision. We presented detailed
numerical illustrations for the real and imaginary parts of the different form
factors.

The present work resulted in a deeper analytic understanding of the massive
three–loop form factors, beyond the numerical solutions given in Ref. [16,
17]. We have compared to the numerical results given in [16, 17] and found
agreement, cf. Section 5 for details. Ref. [16, 17] estimated the numerical
accuracy of the $O(\varepsilon^{0})$ contribution based on that obtained for
the pole terms, which are known analytically. This method is not always
reliable. Ancillary files are provided for analytic and numerical comparisons.
The ancillary files to the present paper are given at [79].

Acknowledgments.
We would like to thank A. Behring and K. Schönwald for discussions. This work
was supported by the Austrian Science Fund (FWF) grant P33530. N.R. has very
essentially contributed to the calculation of the physics amplitude. The
Feynman diagrams have been drawn using Axodraw [80].

##  Appendix A Vector coefficients at $x=0$ in terms of known constants

In this appendix we show the first few coefficients of the expansions at $x=0$
of the vector form factors $F_{v,1,i}^{(0)}(x)$ and $F_{v,2,i}^{(0)}(x)$, with
$i=1,2,3$, as given in Eq. (3.2). The corresponding constants are given in
Eqs. (3.31), (3.32) and (3.33). In other words, we provide the first few
coefficients $c_{I,i}(j,k)$ in Eq. (3.16) for
$I\in\left\\{(v,1),\,(v,2)\right\\}$. The coefficients in the case of other
form factors look similar. The coefficients of $F_{v,1,2}^{(0)}(x)$ are given
by

$\displaystyle c_{v,1,2}(4,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.1)
$\displaystyle c_{v,1,2}(4,-1)$ $\displaystyle=$ $\displaystyle 0,$ (A.2)
$\displaystyle c_{v,1,2}(4,0)$ $\displaystyle=$ $\displaystyle 0,$ (A.3)
$\displaystyle c_{v,1,2}(4,1)$ $\displaystyle=$ $\displaystyle 0,$ (A.4)
$\displaystyle c_{v,1,2}(4,2)$ $\displaystyle=$ $\displaystyle 0,$ (A.5)
$\displaystyle c_{v,1,2}(4,3)$ $\displaystyle=$ $\displaystyle 0,$ (A.6)
$\displaystyle c_{v,1,2}(3,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.7)
$\displaystyle c_{v,1,2}(3,-1)$ $\displaystyle=$ $\displaystyle-128,$ (A.8)
$\displaystyle c_{v,1,2}(3,0)$ $\displaystyle=$ $\displaystyle-\frac{832}{9},$
(A.9) $\displaystyle c_{v,1,2}(3,1)$ $\displaystyle=$ $\displaystyle
256l_{2}+\frac{92416}{81},$ (A.10) $\displaystyle c_{v,1,2}(3,2)$
$\displaystyle=$ $\displaystyle\frac{41344}{27},$ (A.11) $\displaystyle
c_{v,1,2}(3,3)$ $\displaystyle=$ $\displaystyle 256l_{2}+\frac{17920}{81},$
(A.12) $\displaystyle c_{v,1,2}(2,-2)$ $\displaystyle=$ $\displaystyle 0,$
(A.13) $\displaystyle c_{v,1,2}(2,-1)$ $\displaystyle=$
$\displaystyle\frac{512}{9}l_{2}+\frac{18464}{81},$ (A.14) $\displaystyle
c_{v,1,2}(2,0)$ $\displaystyle=$
$\displaystyle-\frac{15616}{9}l_{2}-\frac{97792}{81},$ (A.15) $\displaystyle
c_{v,1,2}(2,1)$ $\displaystyle=$
$\displaystyle-256\zeta_{2}+2048l_{2}-\frac{995296}{81},$ (A.16)
$\displaystyle c_{v,1,2}(2,2)$ $\displaystyle=$
$\displaystyle-\frac{1024}{3}l_{2}+\frac{579520}{27},$ (A.17) $\displaystyle
c_{v,1,2}(2,3)$ $\displaystyle=$
$\displaystyle-256\zeta_{2}-\frac{7168}{9}l_{2}-\frac{2448448}{81},$ (A.18)
$\displaystyle c_{v,1,2}(1,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.19)
$\displaystyle c_{v,1,2}(1,-1)$ $\displaystyle=$
$\displaystyle\frac{14656}{27}\zeta_{2}+\frac{62080}{27}l_{2}-\frac{512}{3}l_{2}^{2}+\frac{125968}{81},$
(A.20) $\displaystyle c_{v,1,2}(1,0)$ $\displaystyle=$
$\displaystyle\frac{256}{9}\zeta_{2}-\frac{432128}{27}l_{2}+\frac{2560}{3}l_{2}^{2}-\frac{870272}{81},$
(A.21) $\displaystyle c_{v,1,2}(1,1)$ $\displaystyle=$
$\displaystyle-\frac{71360}{27}\zeta_{2}+\frac{290944}{27}l_{2}-1536l_{2}^{2}-512\zeta_{3}+\frac{9540560}{243},$
(A.22) $\displaystyle c_{v,1,2}(1,2)$ $\displaystyle=$
$\displaystyle-\frac{207872}{27}\zeta_{2}-\frac{182272}{27}l_{2}+\frac{5120}{3}l_{2}^{2}-\frac{1943552}{27},$
(A.23) $\displaystyle c_{v,1,2}(1,3)$ $\displaystyle=$
$\displaystyle\frac{101888}{27}\zeta_{2}-\frac{131584}{3}l_{2}-\frac{5120}{3}l_{2}^{2}-512\zeta_{3}+\frac{18393728}{243},$
(A.24) $\displaystyle c_{v,1,2}(0,-2)$ $\displaystyle=$
$\displaystyle-\frac{177152}{81}l_{2},$ (A.25) $\displaystyle c_{v,1,2}(0,-1)$
$\displaystyle=$
$\displaystyle\frac{156256}{81}\zeta_{2}+\frac{1429312}{81}l_{2}-\frac{37120}{9}\zeta_{2}l_{2}$
(A.26)
$\displaystyle-\frac{512}{3}l_{2}^{2}-\frac{224}{3}\zeta_{3}+\frac{2731388}{81},$
$\displaystyle c_{v,1,2}(0,0)$ $\displaystyle=$
$\displaystyle\frac{166528}{27}\zeta_{2}-\frac{6199040}{81}l_{2}+\frac{149504}{9}\zeta_{2}l_{2}$
(A.27)
$\displaystyle+\frac{2048}{3}l_{2}^{2}+\frac{9856}{9}\zeta_{3}-\frac{43085848}{243},$
$\displaystyle c_{v,1,2}(0,1)$ $\displaystyle=$
$\displaystyle-\frac{1033504}{27}\zeta_{2}-128\zeta_{2}^{2}+\frac{1066304}{9}l_{2}-31488\zeta_{2}l_{2}$
(A.28)
$\displaystyle-\frac{6656}{3}l_{2}^{2}+\frac{3296}{3}\zeta_{3}+\frac{46115828}{243},$
$\displaystyle c_{v,1,2}(0,2)$ $\displaystyle=$
$\displaystyle\frac{486400}{27}\zeta_{2}-\frac{14123776}{81}l_{2}+\frac{299008}{9}\zeta_{2}l_{2}$
(A.29)
$\displaystyle+4096l_{2}^{2}-\frac{14080}{27}\zeta_{3}-\frac{159872992}{729},$
$\displaystyle c_{v,1,2}(0,3)$ $\displaystyle=$
$\displaystyle-\frac{873344}{27}\zeta_{2}-128\zeta_{2}^{2}+\frac{5127808}{27}l_{2}-\frac{320512}{9}\zeta_{2}l_{2}$
(A.30) $\displaystyle-6144l_{2}^{2}-1024\zeta_{3}+\frac{706205288}{3645}\,.$

The coefficients of $F_{v,2,2}^{(0)}(x)$ read

$\displaystyle c_{v,2,2}(4,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.31)
$\displaystyle c_{v,2,2}(4,-1)$ $\displaystyle=$ $\displaystyle 0,$ (A.32)
$\displaystyle c_{v,2,2}(4,0)$ $\displaystyle=$ $\displaystyle 8,$ (A.33)
$\displaystyle c_{v,2,2}(4,1)$ $\displaystyle=$ $\displaystyle 0,$ (A.34)
$\displaystyle c_{v,2,2}(4,2)$ $\displaystyle=$ $\displaystyle 16,$ (A.35)
$\displaystyle c_{v,2,2}(4,3)$ $\displaystyle=$ $\displaystyle 0,$ (A.36)
$\displaystyle c_{v,2,2}(3,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.37)
$\displaystyle c_{v,2,2}(3,-1)$ $\displaystyle=$ $\displaystyle-64,$ (A.38)
$\displaystyle c_{v,2,2}(3,0)$ $\displaystyle=$ $\displaystyle
128l_{2}+\frac{6592}{27},$ (A.39) $\displaystyle c_{v,2,2}(3,1)$
$\displaystyle=$ $\displaystyle-512l_{2}-\frac{47968}{27},$ (A.40)
$\displaystyle c_{v,2,2}(3,2)$ $\displaystyle=$ $\displaystyle
256l_{2}+\frac{1888}{27},$ (A.41) $\displaystyle c_{v,2,2}(3,3)$
$\displaystyle=$ $\displaystyle-512l_{2}-\frac{79552}{27},$ (A.42)
$\displaystyle c_{v,2,2}(2,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.43)
$\displaystyle c_{v,2,2}(2,-1)$ $\displaystyle=$
$\displaystyle\frac{256}{9}l_{2}-\frac{320}{81},$ (A.44) $\displaystyle
c_{v,2,2}(2,0)$ $\displaystyle=$ $\displaystyle
64\zeta_{2}+\frac{10880}{9}l_{2}+\frac{113536}{81},$ (A.45) $\displaystyle
c_{v,2,2}(2,1)$ $\displaystyle=$
$\displaystyle-64\zeta_{2}-\frac{8576}{9}l_{2}+\frac{183632}{27},$ (A.46)
$\displaystyle c_{v,2,2}(2,2)$ $\displaystyle=$ $\displaystyle
128\zeta_{2}+\frac{896}{9}l_{2}-\frac{176864}{27},$ (A.47) $\displaystyle
c_{v,2,2}(2,3)$ $\displaystyle=$
$\displaystyle-64\zeta_{2}-\frac{22528}{9}l_{2}+\frac{174184}{27},$ (A.48)
$\displaystyle c_{v,2,2}(1,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.49)
$\displaystyle c_{v,2,2}(1,-1)$ $\displaystyle=$
$\displaystyle\frac{7328}{27}\zeta_{2}+\frac{27968}{27}l_{2}-\frac{256}{3}l_{2}^{2}+\frac{1240}{81},$
(A.50) $\displaystyle c_{v,2,2}(1,0)$ $\displaystyle=$
$\displaystyle-\frac{5056}{9}\zeta_{2}+\frac{8960}{3}l_{2}-\frac{512}{3}l_{2}^{2}+\frac{33008}{27},$
(A.51) $\displaystyle c_{v,2,2}(1,1)$ $\displaystyle=$
$\displaystyle\frac{161696}{27}\zeta_{2}+\frac{242368}{27}l_{2}+\frac{512}{3}l_{2}^{2}+128\zeta_{3}-\frac{63064}{27},$
(A.52) $\displaystyle c_{v,2,2}(1,2)$ $\displaystyle=$
$\displaystyle\frac{18560}{27}\zeta_{2}+\frac{63872}{9}l_{2}-\frac{1024}{3}l_{2}^{2}+\frac{861728}{81},$
(A.53) $\displaystyle c_{v,2,2}(1,3)$ $\displaystyle=$
$\displaystyle\frac{203392}{27}\zeta_{2}+\frac{319424}{27}l_{2}+\frac{256}{3}l_{2}^{2}+128\zeta_{3}-\frac{2598124}{243},$
(A.54) $\displaystyle c_{v,2,2}(0,-2)$ $\displaystyle=$
$\displaystyle-\frac{15872}{27}l_{2},$ (A.55) $\displaystyle c_{v,2,2}(0,-1)$
$\displaystyle=$
$\displaystyle\frac{82256}{81}\zeta_{2}+\frac{367808}{81}l_{2}-\frac{4736}{9}\zeta_{2}l_{2}+\frac{64}{3}l_{2}^{2}$
(A.56) $\displaystyle-\frac{592}{3}\zeta_{3}+\frac{3186380}{243},$
$\displaystyle c_{v,2,2}(0,0)$ $\displaystyle=$
$\displaystyle-6560\zeta_{2}+\frac{2112}{5}\zeta_{2}^{2}+\frac{103168}{9}l_{2}+\frac{4352}{9}\zeta_{2}l_{2}$
(A.57) $\displaystyle-\frac{10496}{9}\zeta_{3}+\frac{10523224}{243},$
$\displaystyle c_{v,2,2}(0,1)$ $\displaystyle=$
$\displaystyle\frac{3050672}{81}\zeta_{2}-\frac{3008}{5}\zeta_{2}^{2}+\frac{140480}{27}l_{2}+\frac{16000}{9}\zeta_{2}l_{2}$
(A.58)
$\displaystyle+\frac{64}{3}l_{2}^{2}+\frac{7664}{9}\zeta_{3}+\frac{59828}{243},$
$\displaystyle c_{v,2,2}(0,2)$ $\displaystyle=$
$\displaystyle-\frac{529600}{27}\zeta_{2}+\frac{4224}{5}\zeta_{2}^{2}-\frac{475072}{81}l_{2}+\frac{8704}{9}\zeta_{2}l_{2}$
(A.59) $\displaystyle-\frac{23936}{27}\zeta_{3}-\frac{18467960}{729},$
$\displaystyle c_{v,2,2}(0,3)$ $\displaystyle=$
$\displaystyle\frac{3651488}{81}\zeta_{2}-\frac{3008}{5}\zeta_{2}^{2}-\frac{151184}{27}l_{2}+\frac{11264}{9}\zeta_{2}l_{2}$
(A.60) $\displaystyle+1408\zeta_{3}-\frac{29436937}{1215}\,.$

The first coefficients of $F_{v,1,3}^{(0)}(x)$ are

$\displaystyle c_{v,1,3}(5,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.61)
$\displaystyle c_{v,1,3}(5,-1)$ $\displaystyle=$ $\displaystyle 0,$ (A.62)
$\displaystyle c_{v,1,3}(5,0)$ $\displaystyle=$ $\displaystyle 0,$ (A.63)
$\displaystyle c_{v,1,3}(5,1)$ $\displaystyle=$
$\displaystyle\frac{224}{5\pi^{2}},$ (A.64) $\displaystyle c_{v,1,3}(5,2)$
$\displaystyle=$ $\displaystyle 0,$ (A.65) $\displaystyle c_{v,1,3}(5,3)$
$\displaystyle=$ $\displaystyle\frac{224}{5\pi^{2}},$ (A.66) $\displaystyle
c_{v,1,3}(4,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.67) $\displaystyle
c_{v,1,3}(4,-1)$ $\displaystyle=$ $\displaystyle-\frac{1904}{81\pi^{2}},$
(A.68) $\displaystyle c_{v,1,3}(4,0)$ $\displaystyle=$
$\displaystyle-\frac{3248}{81\pi^{2}},$ (A.69) $\displaystyle c_{v,1,3}(4,1)$
$\displaystyle=$ $\displaystyle\frac{3080}{27\pi^{2}},$ (A.70) $\displaystyle
c_{v,1,3}(4,2)$ $\displaystyle=$ $\displaystyle\frac{52640}{27\pi^{2}},$
(A.71) $\displaystyle c_{v,1,3}(4,3)$ $\displaystyle=$
$\displaystyle\frac{44128}{81\pi^{2}},$ (A.72) $\displaystyle c_{v,1,3}(3,-2)$
$\displaystyle=$ $\displaystyle 0,$ (A.73) $\displaystyle c_{v,1,3}(3,-1)$
$\displaystyle=$ $\displaystyle-\frac{70336}{243\pi^{2}},$ (A.74)
$\displaystyle c_{v,1,3}(3,0)$ $\displaystyle=$
$\displaystyle\frac{250880}{243\pi^{2}},$ (A.75) $\displaystyle
c_{v,1,3}(3,1)$ $\displaystyle=$
$\displaystyle-\frac{512}{9}-\frac{117712}{81\pi^{2}},$ (A.76) $\displaystyle
c_{v,1,3}(3,2)$ $\displaystyle=$ $\displaystyle\frac{5944064}{243\pi^{2}},$
(A.77) $\displaystyle c_{v,1,3}(3,3)$ $\displaystyle=$
$\displaystyle-\frac{512}{9}-\frac{5339824}{243\pi^{2}},$ (A.78)
$\displaystyle c_{v,1,3}(2,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.79)
$\displaystyle c_{v,1,3}(2,-1)$ $\displaystyle=$
$\displaystyle-\frac{17696}{81}-\frac{447104}{243\pi^{2}},$ (A.80)
$\displaystyle c_{v,1,3}(2,0)$ $\displaystyle=$
$\displaystyle\frac{2964416}{243\pi^{2}}-\frac{58240}{81},$ (A.81)
$\displaystyle c_{v,1,3}(2,1)$ $\displaystyle=$ $\displaystyle
896l_{2}+\frac{896}{\pi^{2}}\zeta_{3}+\frac{185888}{81}-\frac{11501560}{243\pi^{2}},$
(A.82) $\displaystyle c_{v,1,3}(2,2)$ $\displaystyle=$
$\displaystyle\frac{143104}{81}+\frac{13919248}{81\pi^{2}},$ (A.83)
$\displaystyle c_{v,1,3}(2,3)$ $\displaystyle=$ $\displaystyle
896l_{2}+\frac{896}{\pi^{2}}\zeta_{3}+\frac{69952}{27}-\frac{28909132}{243\pi^{2}},$
(A.84) $\displaystyle c_{v,1,3}(1,-2)$ $\displaystyle=$ $\displaystyle 0,$
(A.85) $\displaystyle c_{v,1,3}(1,-1)$ $\displaystyle=$
$\displaystyle-\frac{1792}{9}l_{2}+\frac{60032}{27\pi^{2}}\zeta_{3}+\frac{87056}{243}-\frac{1974784}{243\pi^{2}},$
(A.86) $\displaystyle c_{v,1,3}(1,0)$ $\displaystyle=$
$\displaystyle-\frac{7168}{9}l_{2}-\frac{12544}{\pi^{2}}\zeta_{3}+\frac{11072}{9}+\frac{662144}{27\pi^{2}},$
(A.87) $\displaystyle c_{v,1,3}(1,1)$ $\displaystyle=$
$\displaystyle\frac{9856}{5}\zeta_{2}+\frac{490112}{27\pi^{2}}\zeta_{3}-\frac{4687888}{243}-\frac{2968168}{243\pi^{2}},$
(A.88) $\displaystyle c_{v,1,3}(1,2)$ $\displaystyle=$
$\displaystyle\frac{17920}{9}l_{2}-\frac{272384}{9\pi^{2}}\zeta_{3}+\frac{7113728}{243}+\frac{145353152}{729\pi^{2}},$
(A.89) $\displaystyle c_{v,1,3}(1,3)$ $\displaystyle=$
$\displaystyle-\frac{10150048}{243}+\frac{14336l_{2}}{9}-\frac{115553998}{729\pi^{2}}+\frac{9856\zeta_{2}}{5}-\frac{1250816\zeta_{3}}{27\pi^{2}},$
(A.90) $\displaystyle c_{v,1,3}(0,-2)$ $\displaystyle=$
$\displaystyle-\frac{221440}{243},$ (A.91) $\displaystyle c_{v,1,3}(0,-1)$
$\displaystyle=$
$\displaystyle-\frac{14560}{81}\zeta_{2}+\frac{26432}{81}l_{2}+\frac{794752}{81\pi^{2}}\zeta_{3}+\frac{11712400}{729}-\frac{4659760}{243\pi^{2}},$
(A.92) $\displaystyle c_{v,1,3}(0,0)$ $\displaystyle=$
$\displaystyle-\frac{590848}{135}\zeta_{2}-\frac{105728}{27}l_{2}-\frac{160384}{3\pi^{2}}\zeta_{3}+\frac{17780672}{243\pi^{2}}-\frac{71864320}{729},$
(A.93) $\displaystyle c_{v,1,3}(0,1)$ $\displaystyle=$
$\displaystyle\frac{2887936}{405}\zeta_{2}-\frac{688576}{81}l_{2}+3584\zeta_{2}l_{2}+1344\zeta_{3}+\frac{10114048}{81\pi^{2}}\zeta_{3}$
(A.94)
$\displaystyle+\frac{448}{\pi^{2}}\zeta_{5}+\frac{120607088}{729}-\frac{3788428}{27\pi^{2}},$
$\displaystyle c_{v,1,3}(0,2)$ $\displaystyle=$
$\displaystyle-\frac{1532672}{405}\zeta_{2}-\frac{476672}{81}l_{2}-\frac{8866816}{81\pi^{2}}\zeta_{3}+\frac{653277352}{2187\pi^{2}}$
(A.95) $\displaystyle-\frac{192621760}{729},$ $\displaystyle c_{v,1,3}(0,3)$
$\displaystyle=$
$\displaystyle-\frac{2014528}{405}\zeta_{2}+\frac{68992}{3}l_{2}+3584\zeta_{2}l_{2}+1344\zeta_{3}-\frac{905408}{81\pi^{2}}\zeta_{3}$
(A.96)
$\displaystyle+\frac{448}{\pi^{2}}\zeta_{5}+\frac{194264248}{729}-\frac{1214278177}{4374\pi^{2}},$

and likewise for $F_{v,2,3}^{(0)}(x)$

$\displaystyle c_{v,2,3}(5,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.97)
$\displaystyle c_{v,2,3}(5,-1)$ $\displaystyle=$ $\displaystyle 0,$ (A.98)
$\displaystyle c_{v,2,3}(5,0)$ $\displaystyle=$
$\displaystyle-\frac{112}{15\pi^{2}},$ (A.99) $\displaystyle c_{v,2,3}(5,1)$
$\displaystyle=$ $\displaystyle\frac{476}{15\pi^{2}},$ (A.100) $\displaystyle
c_{v,2,3}(5,2)$ $\displaystyle=$ $\displaystyle-\frac{224}{15\pi^{2}},$
(A.101) $\displaystyle c_{v,2,3}(5,3)$ $\displaystyle=$
$\displaystyle\frac{476}{15\pi^{2}},$ (A.102) $\displaystyle c_{v,2,3}(4,-2)$
$\displaystyle=$ $\displaystyle 0,$ (A.103) $\displaystyle c_{v,2,3}(4,-1)$
$\displaystyle=$ $\displaystyle-\frac{952}{81\pi^{2}},$ (A.104) $\displaystyle
c_{v,2,3}(4,0)$ $\displaystyle=$ $\displaystyle\frac{6944}{81\pi^{2}},$
(A.105) $\displaystyle c_{v,2,3}(4,1)$ $\displaystyle=$
$\displaystyle-\frac{14056}{27\pi^{2}},$ (A.106) $\displaystyle
c_{v,2,3}(4,2)$ $\displaystyle=$ $\displaystyle-\frac{728}{3\pi^{2}},$ (A.107)
$\displaystyle c_{v,2,3}(4,3)$ $\displaystyle=$
$\displaystyle-\frac{162400}{81\pi^{2}},$ (A.108) $\displaystyle
c_{v,2,3}(3,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.109) $\displaystyle
c_{v,2,3}(3,-1)$ $\displaystyle=$ $\displaystyle-\frac{39200}{243\pi^{2}},$
(A.110) $\displaystyle c_{v,2,3}(3,0)$ $\displaystyle=$
$\displaystyle\frac{320}{9}+\frac{7616}{81\pi^{2}},$ (A.111) $\displaystyle
c_{v,2,3}(3,1)$ $\displaystyle=$
$\displaystyle-\frac{1312}{9}-\frac{941248}{243\pi^{2}},$ (A.112)
$\displaystyle c_{v,2,3}(3,2)$ $\displaystyle=$
$\displaystyle\frac{640}{9}-\frac{790216}{243\pi^{2}},$ (A.113) $\displaystyle
c_{v,2,3}(3,3)$ $\displaystyle=$
$\displaystyle-\frac{1312}{9}-\frac{176512}{27\pi^{2}},$ (A.114)
$\displaystyle c_{v,2,3}(2,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.115)
$\displaystyle c_{v,2,3}(2,-1)$ $\displaystyle=$
$\displaystyle-\frac{8848}{81}-\frac{322336}{243\pi^{2}},$ (A.116)
$\displaystyle c_{v,2,3}(2,0)$ $\displaystyle=$
$\displaystyle\frac{32560}{81}-448l_{2}+\frac{677152}{243\pi^{2}}+\frac{896\zeta_{3}}{\pi^{2}},$
(A.117) $\displaystyle c_{v,2,3}(2,1)$ $\displaystyle=$
$\displaystyle-\frac{273584}{81}-\frac{3312008}{243\pi^{2}}-\frac{2688\zeta_{3}}{\pi^{2}}$
(A.118) $\displaystyle c_{v,2,3}(2,2)$ $\displaystyle=$
$\displaystyle-\frac{70624}{81}-896l_{2}-\frac{2675876}{81\pi^{2}}+\frac{1792\zeta_{3}}{\pi^{2}},$
(A.119) $\displaystyle c_{v,2,3}(2,3)$ $\displaystyle=$
$\displaystyle-\frac{183808}{27}-\frac{1477532}{27\pi^{2}}-\frac{2688\zeta_{3}}{\pi^{2}},$
(A.120) $\displaystyle c_{v,2,3}(1,-2)$ $\displaystyle=$ $\displaystyle 0,$
(A.121) $\displaystyle c_{v,2,3}(1,-1)$ $\displaystyle=$
$\displaystyle-\frac{126008}{243}-\frac{896l_{2}}{9}-\frac{1073632}{243\pi^{2}}+\frac{30016\zeta_{3}}{27\pi^{2}},$
(A.122) $\displaystyle c_{v,2,3}(1,0)$ $\displaystyle=$
$\displaystyle\frac{130192}{243}-\frac{5824l_{2}}{9}-\frac{801472}{81\pi^{2}}+64\zeta_{2}+\frac{25088\zeta_{3}}{3\pi^{2}},$
(A.123) $\displaystyle c_{v,2,3}(1,1)$ $\displaystyle=$
$\displaystyle\frac{1937720}{243}-\frac{30464l_{2}}{9}-\frac{13104616}{243\pi^{2}}-\frac{27232\zeta_{2}}{15}+\frac{169792\zeta_{3}}{27\pi^{2}},$
(A.124) $\displaystyle c_{v,2,3}(1,2)$ $\displaystyle=$
$\displaystyle-\frac{423584}{81}-\frac{19712l_{2}}{9}-\frac{36706208}{729\pi^{2}}+128\zeta_{2}+\frac{186368\zeta_{3}}{9\pi^{2}},$
(A.125) $\displaystyle c_{v,2,3}(1,3)$ $\displaystyle=$
$\displaystyle\frac{2046752}{243}-\frac{52864l_{2}}{9}-\frac{54828256}{729\pi^{2}}-\frac{27232\zeta_{2}}{15}+\frac{982016\zeta_{3}}{27\pi^{2}},$
(A.126) $\displaystyle c_{v,2,3}(0,-2)$ $\displaystyle=$
$\displaystyle-\frac{19840}{81},$ (A.127) $\displaystyle c_{v,2,3}(0,-1)$
$\displaystyle=$
$\displaystyle\frac{4465256}{729}+\frac{38752l_{2}}{81}-\frac{1908592}{243\pi^{2}}+\frac{124304\zeta_{2}}{405}+\frac{85568\zeta_{3}}{81\pi^{2}},$
(A.128) $\displaystyle c_{v,2,3}(0,0)$ $\displaystyle=$
$\displaystyle\frac{13510576}{729}+\frac{72128l_{2}}{81}-\frac{2291072}{243\pi^{2}}+\frac{549968\zeta_{2}}{135}-1792l_{2}\zeta_{2}-\frac{2432\zeta_{3}}{3}$
(A.129)
$\displaystyle+\frac{14336\zeta_{3}}{81\pi^{2}}+\frac{10752\zeta_{5}}{\pi^{2}},$
$\displaystyle c_{v,2,3}(0,1)$ $\displaystyle=$
$\displaystyle\frac{3356632}{729}+7392l_{2}+\frac{1120700}{243\pi^{2}}-\frac{2580928\zeta_{2}}{405}-\frac{1184\zeta_{3}}{3}-\frac{2202368\zeta_{3}}{81\pi^{2}}$
(A.130) $\displaystyle-\frac{6944\zeta_{5}}{\pi^{2}},$ $\displaystyle
c_{v,2,3}(0,2)$ $\displaystyle=$
$\displaystyle\frac{5003536}{729}-\frac{26432l_{2}}{9}-\frac{110330920}{2187\pi^{2}}+\frac{2573792\zeta_{2}}{405}-3584l_{2}\zeta_{2}-\frac{4864\zeta_{3}}{3}$
(A.131)
$\displaystyle+\frac{674240\zeta_{3}}{81\pi^{2}}+\frac{21504\zeta_{5}}{\pi^{2}},$
$\displaystyle c_{v,2,3}(0,3)$ $\displaystyle=$
$\displaystyle-\frac{4741880}{729}+\frac{363776l_{2}}{81}+\frac{102560675}{4374\pi^{2}}-\frac{1971968\zeta_{2}}{405}-\frac{1184\zeta_{3}}{3}$
(A.132)
$\displaystyle+\frac{151648\zeta_{3}}{81\pi^{2}}-\frac{6944\zeta_{5}}{\pi^{2}}.$

The coefficients of $F_{v,1,1}^{(0)}(x)$ are

$\displaystyle c_{v,1,1}(6,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.133)
$\displaystyle c_{v,1,1}(6,-1)$ $\displaystyle=$ $\displaystyle 0,$ (A.134)
$\displaystyle c_{v,1,1}(6,0)$ $\displaystyle=$ $\displaystyle 0,$ (A.135)
$\displaystyle c_{v,1,1}(6,1)$ $\displaystyle=$ $\displaystyle 0,$ (A.136)
$\displaystyle c_{v,1,1}(6,2)$ $\displaystyle=$ $\displaystyle 0,$ (A.137)
$\displaystyle c_{v,1,1}(6,3)$ $\displaystyle=$ $\displaystyle 0,$ (A.138)
$\displaystyle c_{v,1,1}(5,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.139)
$\displaystyle c_{v,1,1}(5,-1)$ $\displaystyle=$
$\displaystyle-\frac{704}{81},$ (A.140) $\displaystyle c_{v,1,1}(5,0)$
$\displaystyle=$ $\displaystyle\frac{13904}{405},$ (A.141) $\displaystyle
c_{v,1,1}(5,1)$ $\displaystyle=$
$\displaystyle-\frac{88}{45}-\frac{224}{5}\frac{\zeta_{3}}{\pi^{2}},$ (A.142)
$\displaystyle c_{v,1,1}(5,2)$ $\displaystyle=$
$\displaystyle\frac{6944}{81},$ (A.143) $\displaystyle c_{v,1,1}(5,3)$
$\displaystyle=$
$\displaystyle-\frac{22384}{405}-\frac{224\zeta_{3}}{5\pi^{2}},$ (A.144)
$\displaystyle c_{v,1,1}(4,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.145)
$\displaystyle c_{v,1,1}(4,-1)$ $\displaystyle=$
$\displaystyle-\frac{688}{243}+\frac{1904}{81}\frac{\zeta_{3}}{\pi^{2}},$
(A.146) $\displaystyle c_{v,1,1}(4,0)$ $\displaystyle=$
$\displaystyle\frac{6944}{81}+\frac{3248}{81}\frac{\zeta_{3}}{\pi^{2}},$
(A.147) $\displaystyle c_{v,1,1}(4,1)$ $\displaystyle=$
$\displaystyle-\frac{193676}{243}+16\zeta_{2}-\frac{3080}{27}\frac{\zeta_{3}}{\pi^{2}},$
(A.148) $\displaystyle c_{v,1,1}(4,2)$ $\displaystyle=$
$\displaystyle\frac{63224}{81}-\frac{52640}{27}\frac{\zeta_{3}}{\pi^{2}},$
(A.149) $\displaystyle c_{v,1,1}(4,3)$ $\displaystyle=$
$\displaystyle-\frac{190484}{81}+16\zeta_{2}-\frac{44128}{81}\frac{\zeta_{3}}{\pi^{2}},$
(A.150) $\displaystyle c_{v,1,1}(3,-2)$ $\displaystyle=$ $\displaystyle 0,$
(A.151) $\displaystyle c_{v,1,1}(3,-1)$ $\displaystyle=$ $\displaystyle
240+\frac{256}{3}\zeta_{2}+\frac{70336}{243}\frac{\zeta_{3}}{\pi^{2}},$
(A.152) $\displaystyle c_{v,1,1}(3,0)$ $\displaystyle=$
$\displaystyle-\frac{443584}{243}-\frac{7936}{81}\zeta_{2}-\frac{250880}{243}\frac{\zeta_{3}}{\pi^{2}},$
(A.153) $\displaystyle c_{v,1,1}(3,1)$ $\displaystyle=$
$\displaystyle\frac{4446616}{729}-\frac{16832}{81}\zeta_{2}+\frac{512}{9}\zeta_{3}+\frac{117712}{81}\frac{\zeta_{3}}{\pi^{2}},$
(A.154) $\displaystyle c_{v,1,1}(3,2)$ $\displaystyle=$
$\displaystyle-\frac{4453616}{243}-\frac{8576}{9}\zeta_{2}-\frac{5944064}{243}\frac{\zeta_{3}}{\pi^{2}},$
(A.155) $\displaystyle c_{v,1,1}(3,3)$ $\displaystyle=$
$\displaystyle\frac{15976564}{729}+\frac{5632}{81}\zeta_{2}+\frac{512}{9}\zeta_{3}+\frac{5339824}{243}\frac{\zeta_{3}}{\pi^{2}},$
(A.156) $\displaystyle c_{v,1,1}(2,-2)$ $\displaystyle=$ $\displaystyle 0,$
(A.157) $\displaystyle c_{v,1,1}(2,-1)$ $\displaystyle=$
$\displaystyle\frac{73096}{27}-\frac{3136}{27}\zeta_{2}-\frac{736}{81}\zeta_{3}+\frac{447104}{243}\frac{\zeta_{3}}{\pi^{2}},$
(A.158) $\displaystyle c_{v,1,1}(2,0)$ $\displaystyle=$
$\displaystyle-\frac{3005056}{243}+\frac{13952}{27}\zeta_{2}+\frac{194752}{81}\zeta_{3}-\frac{2964416}{243}\frac{\zeta_{3}}{\pi^{2}},$
(A.159) $\displaystyle c_{v,1,1}(2,1)$ $\displaystyle=$
$\displaystyle\frac{6205580}{729}+\frac{157280}{27}\zeta_{2}+\frac{1184}{5}\zeta_{2}^{2}+256\zeta_{2}l_{2}^{2}-\frac{128}{3}l_{2}^{4}$
(A.160)
$\displaystyle-1024a_{4}-\frac{533408}{81}\zeta_{3}+\frac{11501560}{243}\frac{\zeta_{3}}{\pi^{2}}-896\zeta_{3}l_{2}-896\frac{\zeta_{3}^{2}}{\pi^{2}},$
$\displaystyle c_{v,1,1}(2,2)$ $\displaystyle=$
$\displaystyle-\frac{25654520}{729}-\frac{22144}{3}\zeta_{2}+\frac{353408}{81}\zeta_{3}-\frac{13919248}{81}\frac{\zeta_{3}}{\pi^{2}},$
(A.161) $\displaystyle c_{v,1,1}(2,3)$ $\displaystyle=$
$\displaystyle\frac{10452319}{1215}+\frac{312752}{27}\zeta_{2}+\frac{1184}{5}\zeta_{2}^{2}+256\zeta_{2}l_{2}^{2}-\frac{128}{3}l_{2}^{4}$
(A.162)
$\displaystyle-1024a_{4}-\frac{151616}{27}\zeta_{3}+\frac{28909132}{243}\frac{\zeta_{3}}{\pi^{2}}-896\zeta_{3}l_{2}-896\frac{\zeta_{3}^{2}}{\pi^{2}},$
$\displaystyle c_{v,1,1}(1,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.163)
$\displaystyle c_{v,1,1}(1,-1)$ $\displaystyle=$
$\displaystyle\frac{2681840}{243}-\frac{239264}{243}\zeta_{2}+\frac{4544}{45}\zeta_{2}^{2}-\frac{512}{9}\zeta_{2}l_{2}^{2}+\frac{256}{27}l_{2}^{4}$
(A.164)
$\displaystyle+\frac{2048}{9}a_{4}-\frac{755168}{243}\zeta_{3}+\frac{1974784}{243}\frac{\zeta_{3}}{\pi^{2}}+\frac{1792}{9}\zeta_{3}l_{2}-\frac{60032}{27}\frac{\zeta_{3}^{2}}{\pi^{2}},$
$\displaystyle c_{v,1,1}(1,0)$ $\displaystyle=$
$\displaystyle-\frac{11847904}{243}+\frac{1762496}{243}\zeta_{2}-\frac{13952}{135}\zeta_{2}^{2}-\frac{2048}{9}\zeta_{2}l_{2}^{2}+\frac{1024}{27}l_{2}^{4}$
(A.165)
$\displaystyle+\frac{8192}{9}a_{4}+\frac{960512}{81}\zeta_{3}-\frac{662144}{27}\frac{\zeta_{3}}{\pi^{2}}+\frac{7168}{9}\zeta_{3}l_{2}+12544\frac{\zeta_{3}^{2}}{\pi^{2}},$
$\displaystyle c_{v,1,1}(1,1)$ $\displaystyle=$
$\displaystyle\frac{16988900}{2187}-\frac{4210432}{243}\zeta_{2}-\frac{6016}{3}\zeta_{2}^{2}+\frac{6957088}{243}\zeta_{3}$
(A.166)
$\displaystyle+\frac{2968168}{243}\frac{\zeta_{3}}{\pi^{2}}-\frac{9856}{5}\zeta_{2}\zeta_{3}-\frac{490112}{27}\frac{\zeta_{3}^{2}}{\pi^{2}}+1280\zeta_{5},$
$\displaystyle c_{v,1,1}(1,2)$ $\displaystyle=$
$\displaystyle-\frac{23135480}{243}+\frac{9932032}{243}\zeta_{2}-\frac{447872}{135}\zeta_{2}^{2}+\frac{5120}{9}\zeta_{2}l_{2}^{2}$
(A.167)
$\displaystyle-\frac{2560}{27}l_{2}^{4}-\frac{20480}{9}a_{4}-\frac{3832448}{243}\zeta_{3}-\frac{145353152}{729}\frac{\zeta_{3}}{\pi^{2}}$
$\displaystyle-\frac{17920}{9}\zeta_{3}l_{2}+\frac{272384}{9}\frac{\zeta_{3}^{2}}{\pi^{2}},$
$\displaystyle c_{v,1,1}(1,3)$ $\displaystyle=$
$\displaystyle\frac{2962613437}{36450}-\frac{4402552}{81}\zeta_{2}+\frac{51584}{45}\zeta_{2}^{2}+\frac{4096}{9}\zeta_{2}l_{2}^{2}$
(A.168)
$\displaystyle-\frac{2048}{27}l_{2}^{4}-\frac{16384}{9}a_{4}+\frac{15899680}{243}\zeta_{3}+\frac{115553998}{729}\frac{\zeta_{3}}{\pi^{2}}$
$\displaystyle-\frac{9856}{5}\zeta_{2}\zeta_{3}-\frac{14336}{9}\zeta_{3}l_{2}+\frac{1250816}{27}\frac{\zeta_{3}^{2}}{\pi^{2}}+1280\zeta_{5},$
$\displaystyle c_{v,1,1}(0,-2)$ $\displaystyle=$
$\displaystyle\frac{221440}{243}\zeta_{3},$ (A.169) $\displaystyle
c_{v,1,1}(0,-1)$ $\displaystyle=$
$\displaystyle-\frac{28904}{9}-\frac{743888}{243}\zeta_{2}-\frac{95440}{81}\zeta_{2}^{2}+\frac{256}{9}\zeta_{2}^{2}l_{2}+\frac{9344}{81}\zeta_{2}l_{2}^{2}$
(A.170)
$\displaystyle-\frac{4672}{243}l_{2}^{4}-\frac{37376}{81}a_{4}-\frac{10044736}{729}\zeta_{3}+\frac{4659760}{243}\frac{\zeta_{3}}{\pi^{2}}$
$\displaystyle-\frac{38432}{81}\zeta_{2}\zeta_{3}-\frac{26432}{81}\zeta_{3}l_{2}-\frac{794752}{81}\frac{\zeta_{3}^{2}}{\pi^{2}}+\frac{15856}{9}\zeta_{5},$
$\displaystyle c_{v,1,1}(0,0)$ $\displaystyle=$
$\displaystyle\frac{10612672}{243}+\frac{3300608}{243}\zeta_{2}+\frac{349888}{135}\zeta_{2}^{2}-\frac{1024}{9}\zeta_{2}^{2}l_{2}-\frac{268480}{9}\zeta_{5}$
(A.171)
$\displaystyle-\frac{25088}{27}\zeta_{2}l_{2}^{2}+\frac{12544}{81}l_{2}^{4}+\frac{100352}{27}a_{4}+\frac{46482880}{729}\zeta_{3}$
$\displaystyle-\frac{17780672}{243}\frac{\zeta_{3}}{\pi^{2}}+\frac{1078528}{135}\zeta_{2}\zeta_{3}+\frac{105728}{27}\zeta_{3}l_{2}+\frac{160384}{3}\frac{\zeta_{3}^{2}}{\pi^{2}},$
$\displaystyle c_{v,1,1}(0,1)$ $\displaystyle=$
$\displaystyle-\frac{641594606}{6561}+\frac{295424}{81}\zeta_{2}+\frac{5199536}{405}\zeta_{2}^{2}+\frac{63296}{35}\zeta_{2}^{3}$
(A.172)
$\displaystyle-\frac{256}{9}\zeta_{2}^{2}l_{2}-\frac{343936}{81}\zeta_{2}l_{2}^{2}+1024\zeta_{2}^{2}l_{2}^{2}-448\frac{\zeta_{3}\zeta_{5}}{\pi^{2}}$
$\displaystyle+\frac{171968}{243}l_{2}^{4}-\frac{512}{3}\zeta_{2}l_{2}^{4}+\frac{1375744}{81}a_{4}+\frac{1642288}{27}\zeta_{5}$
$\displaystyle-4096\zeta_{2}a_{4}-\frac{82803488}{729}\zeta_{3}+\frac{3788428}{27}\frac{\zeta_{3}}{\pi^{2}}-\frac{5779456}{405}\zeta_{2}\zeta_{3}$
$\displaystyle+\frac{688576}{81}\zeta_{3}l_{2}-3584\zeta_{2}\zeta_{3}l_{2}-1024\zeta_{3}^{2}-\frac{10114048}{81}\frac{\zeta_{3}^{2}}{\pi^{2}},$
$\displaystyle c_{v,1,1}(0,2)$ $\displaystyle=$
$\displaystyle\frac{285175364}{6561}+\frac{1200128a_{4}}{81}+\frac{150016l_{2}^{4}}{243}+\frac{5818816\zeta_{2}}{243}-\frac{300032}{81}l_{2}^{2}\zeta_{2}$
(A.173)
$\displaystyle-\frac{7099136\zeta_{2}^{2}}{405}-\frac{2048}{9}l_{2}\zeta_{2}^{2}+\frac{186528256\zeta_{3}}{729}+\frac{476672l_{2}\zeta_{3}}{81}-\frac{653277352\zeta_{3}}{2187\pi^{2}}$
$\displaystyle+\frac{4719872\zeta_{2}\zeta_{3}}{405}+\frac{8866816\zeta_{3}^{2}}{81\pi^{2}}-\frac{653696\zeta_{5}}{9},$
$\displaystyle c_{v,1,1}(0,3)$ $\displaystyle=$
$\displaystyle-\frac{1906863860207}{6561000}-\frac{185344a_{4}}{3}-\frac{23168l_{2}^{4}}{9}-\frac{15147034\zeta_{2}}{729}-4096a_{4}\zeta_{2}$
(A.174)
$\displaystyle+\frac{46336l_{2}^{2}\zeta_{2}}{3}-\frac{512}{3}l_{2}^{4}\zeta_{2}+\frac{8162192\zeta_{2}^{2}}{135}+1024l_{2}^{2}\zeta_{2}^{2}+\frac{63296\zeta_{2}^{3}}{35}$
$\displaystyle-\frac{169868872\zeta_{3}}{729}-\frac{68992l_{2}\zeta_{3}}{3}+\frac{1214278177\zeta_{3}}{4374\pi^{2}}-\frac{1337792}{405}\zeta_{2}\zeta_{3}$
$\displaystyle-3584l_{2}\zeta_{2}\zeta_{3}-1024\zeta_{3}^{2}+\frac{905408\zeta_{3}^{2}}{81\pi^{2}}+\frac{1681792\zeta_{5}}{27}-\frac{448\zeta_{3}\zeta_{5}}{\pi^{2}},$

and those of $F_{v,2,1}^{(0)}(x)$

$\displaystyle c_{v,2,1}(6,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.175)
$\displaystyle c_{v,2,1}(6,-1)$ $\displaystyle=$ $\displaystyle 0,$ (A.176)
$\displaystyle c_{v,2,1}(6,0)$ $\displaystyle=$ $\displaystyle 0,$ (A.177)
$\displaystyle c_{v,2,1}(6,1)$ $\displaystyle=$ $\displaystyle-\frac{34}{45},$
(A.178) $\displaystyle c_{v,2,1}(6,2)$ $\displaystyle=$ $\displaystyle 0,$
(A.179) $\displaystyle c_{v,2,1}(6,3)$ $\displaystyle=$
$\displaystyle-\frac{34}{45},$ (A.180) $\displaystyle c_{v,2,1}(5,-2)$
$\displaystyle=$ $\displaystyle 0,$ (A.181) $\displaystyle c_{v,2,1}(5,-1)$
$\displaystyle=$ $\displaystyle-\frac{352}{81},$ (A.182) $\displaystyle
c_{v,2,1}(5,0)$ $\displaystyle=$
$\displaystyle-\frac{704}{81}+\frac{112}{15}\frac{\zeta_{3}}{\pi^{2}},$
(A.183) $\displaystyle c_{v,2,1}(5,1)$ $\displaystyle=$
$\displaystyle-\frac{17576}{405}-\frac{476}{15}\frac{\zeta_{3}}{\pi^{2}},$
(A.184) $\displaystyle c_{v,2,1}(5,2)$ $\displaystyle=$
$\displaystyle-\frac{6824}{405}+\frac{224}{15}\frac{\zeta_{3}}{\pi^{2}},$
(A.185) $\displaystyle c_{v,2,1}(5,3)$ $\displaystyle=$
$\displaystyle-\frac{31456}{405}-\frac{476}{15}\frac{\zeta_{3}}{\pi^{2}},$
(A.186) $\displaystyle c_{v,2,1}(4,-2)$ $\displaystyle=$ $\displaystyle 0,$
(A.187) $\displaystyle c_{v,2,1}(4,-1)$ $\displaystyle=$
$\displaystyle\frac{976}{243}+\frac{952}{81}\frac{\zeta_{3}}{\pi^{2}},$
(A.188) $\displaystyle c_{v,2,1}(4,0)$ $\displaystyle=$
$\displaystyle\frac{4784}{243}-8\zeta_{2}-\frac{6944}{81}\frac{\zeta_{3}}{\pi^{2}},$
(A.189) $\displaystyle c_{v,2,1}(4,1)$ $\displaystyle=$
$\displaystyle\frac{29620}{81}+\frac{14056}{27}\frac{\zeta_{3}}{\pi^{2}},$
(A.190) $\displaystyle c_{v,2,1}(4,2)$ $\displaystyle=$
$\displaystyle\frac{29512}{243}-16\zeta_{2}+\frac{728}{3}\frac{\zeta_{3}}{\pi^{2}},$
(A.191) $\displaystyle c_{v,2,1}(4,3)$ $\displaystyle=$
$\displaystyle\frac{113356}{243}+\frac{162400}{81}\frac{\zeta_{3}}{\pi^{2}},$
(A.192) $\displaystyle c_{v,2,1}(3,-2)$ $\displaystyle=$ $\displaystyle 0,$
(A.193) $\displaystyle c_{v,2,1}(3,-1)$ $\displaystyle=$
$\displaystyle\frac{5576}{27}+\frac{128}{3}\zeta_{2}+\frac{39200}{243}\frac{\zeta_{3}}{\pi^{2}},$
(A.194) $\displaystyle c_{v,2,1}(3,0)$ $\displaystyle=$
$\displaystyle-\frac{31984}{81}-\frac{1888}{27}\zeta_{2}-\frac{224}{9}\zeta_{3}-\frac{7616}{81}\frac{\zeta_{3}}{\pi^{2}},$
(A.195) $\displaystyle c_{v,2,1}(3,1)$ $\displaystyle=$
$\displaystyle\frac{199544}{81}+\frac{58528}{81}\zeta_{2}+\frac{928}{9}\zeta_{3}+\frac{941248}{243}\frac{\zeta_{3}}{\pi^{2}},$
(A.196) $\displaystyle c_{v,2,1}(3,2)$ $\displaystyle=$
$\displaystyle-\frac{26168}{27}+\frac{9184}{81}\zeta_{2}-\frac{448}{9}\zeta_{3}+\frac{790216}{243}\frac{\zeta_{3}}{\pi^{2}},$
(A.197) $\displaystyle c_{v,2,1}(3,3)$ $\displaystyle=$
$\displaystyle\frac{1605320}{729}+\frac{101152}{81}\zeta_{2}+\frac{928}{9}\zeta_{3}+\frac{176512}{27}\frac{\zeta_{3}}{\pi^{2}},$
(A.198) $\displaystyle c_{v,2,1}(2,-2)$ $\displaystyle=$ $\displaystyle 0,$
(A.199) $\displaystyle c_{v,2,1}(2,-1)$ $\displaystyle=$
$\displaystyle\frac{43232}{27}-\frac{944}{27}\zeta_{2}-\frac{368}{81}\zeta_{3}+\frac{322336}{243}\frac{\zeta_{3}}{\pi^{2}},$
(A.200) $\displaystyle c_{v,2,1}(2,0)$ $\displaystyle=$
$\displaystyle\frac{130480}{27}-\frac{17872}{27}\zeta_{2}-\frac{816}{5}\zeta_{2}^{2}-128\zeta_{2}l_{2}^{2}+\frac{64}{3}l_{2}^{4}$
(A.201)
$\displaystyle+512a_{4}-\frac{92944}{81}\zeta_{3}-\frac{677152}{243}\frac{\zeta_{3}}{\pi^{2}}+448\zeta_{3}l_{2}-896\frac{\zeta_{3}^{2}}{\pi^{2}},$
$\displaystyle c_{v,2,1}(2,1)$ $\displaystyle=$
$\displaystyle\frac{875360}{243}-\frac{86704}{27}\zeta_{2}+\frac{544}{5}\zeta_{2}^{2}+\frac{375056}{81}\zeta_{3}+\frac{3312008}{243}\frac{\zeta_{3}}{\pi^{2}}$
(A.202) $\displaystyle+2688\frac{\zeta_{3}^{2}}{\pi^{2}},$ $\displaystyle
c_{v,2,1}(2,2)$ $\displaystyle=$
$\displaystyle\frac{12501296}{729}+\frac{69008}{27}\zeta_{2}-\frac{1632}{5}\zeta_{2}^{2}-256\zeta_{2}l_{2}^{2}+\frac{128}{3}l_{2}^{4}$
(A.203)
$\displaystyle+1024a_{4}-\frac{160448}{81}\zeta_{3}+\frac{2675876}{81}\frac{\zeta_{3}}{\pi^{2}}+896\zeta_{3}l_{2}$
$\displaystyle-1792\frac{\zeta_{3}^{2}}{\pi^{2}},$ $\displaystyle
c_{v,2,1}(2,3)$ $\displaystyle=$
$\displaystyle-\frac{9839219}{3645}-\frac{106528}{27}\zeta_{2}+\frac{544}{5}\zeta_{2}^{2}+\frac{51712}{9}\zeta_{3}+\frac{1477532}{27}\frac{\zeta_{3}}{\pi^{2}}$
(A.204) $\displaystyle+2688\frac{\zeta_{3}^{2}}{\pi^{2}},$ $\displaystyle
c_{v,2,1}(1,-2)$ $\displaystyle=$ $\displaystyle 0,$ (A.205) $\displaystyle
c_{v,2,1}(1,-1)$ $\displaystyle=$
$\displaystyle\frac{1469552}{243}-\frac{128368}{243}\zeta_{2}+\frac{2272}{45}\zeta_{2}^{2}-\frac{256}{9}\zeta_{2}l_{2}^{2}+\frac{128}{27}l_{2}^{4}$
(A.206)
$\displaystyle+\frac{1024}{9}a_{4}-\frac{315184}{243}\zeta_{3}+\frac{1073632}{243}\frac{\zeta_{3}}{\pi^{2}}+\frac{896}{9}\zeta_{3}l_{2}$
$\displaystyle-\frac{30016}{27}\frac{\zeta_{3}^{2}}{\pi^{2}},$ $\displaystyle
c_{v,2,1}(1,0)$ $\displaystyle=$
$\displaystyle\frac{4850800}{243}-\frac{242336}{243}\zeta_{2}-\frac{65056}{135}\zeta_{2}^{2}-\frac{1664}{9}\zeta_{2}l_{2}^{2}+\frac{832}{27}l_{2}^{4}$
(A.207)
$\displaystyle+\frac{6656}{9}a_{4}-\frac{804832}{243}\zeta_{3}+\frac{801472}{81}\frac{\zeta_{3}}{\pi^{2}}-64\zeta_{2}\zeta_{3}$
$\displaystyle+\frac{5824}{9}\zeta_{3}l_{2}-\frac{25088}{3}\frac{\zeta_{3}^{2}}{\pi^{2}}+256\zeta_{5},$
$\displaystyle c_{v,2,1}(1,1)$ $\displaystyle=$
$\displaystyle\frac{32303216}{729}+\frac{24592}{243}\zeta_{2}+\frac{261568}{135}\zeta_{2}^{2}-\frac{8704}{9}\zeta_{2}l_{2}^{2}$
(A.208)
$\displaystyle+\frac{4352}{27}l_{2}^{4}+\frac{34816}{9}a_{4}-\frac{7996304}{243}\zeta_{3}+\frac{13104616}{243}\frac{\zeta_{3}}{\pi^{2}}$
$\displaystyle+\frac{27232}{15}\zeta_{2}\zeta_{3}+\frac{30464}{9}\zeta_{3}l_{2}-\frac{169792}{27}\frac{\zeta_{3}^{2}}{\pi^{2}}-2880\zeta_{5},$
$\displaystyle c_{v,2,1}(1,2)$ $\displaystyle=$
$\displaystyle-\frac{3166864}{243}-\frac{1317808}{243}\zeta_{2}-\frac{21664}{135}\zeta_{2}^{2}-\frac{5632}{9}\zeta_{2}l_{2}^{2}$
(A.209)
$\displaystyle+\frac{2816}{27}l_{2}^{4}+\frac{22528}{9}a_{4}+\frac{1241824}{81}\zeta_{3}+\frac{36706208}{729}\frac{\zeta_{3}}{\pi^{2}}$
$\displaystyle-128\zeta_{2}\zeta_{3}+\frac{19712}{9}\zeta_{3}l_{2}-\frac{186368}{9}\frac{\zeta_{3}^{2}}{\pi^{2}}+512\zeta_{5},$
$\displaystyle c_{v,2,1}(1,3)$ $\displaystyle=$
$\displaystyle\frac{2238816019}{54675}+\frac{397808}{81}\zeta_{2}+\frac{607072}{135}\zeta_{2}^{2}-\frac{15104}{9}\zeta_{2}l_{2}^{2}$
(A.210)
$\displaystyle+\frac{7552}{27}l_{2}^{4}+\frac{60416}{9}a_{4}-\frac{11472656}{243}\zeta_{3}+\frac{54828256}{729}\frac{\zeta_{3}}{\pi^{2}}$
$\displaystyle+\frac{27232}{15}\zeta_{2}\zeta_{3}+\frac{52864}{9}\zeta_{3}l_{2}-\frac{982016}{27}\frac{\zeta_{3}^{2}}{\pi^{2}}-2880\zeta_{5},$
$\displaystyle c_{v,2,1}(0,-2)$ $\displaystyle=$
$\displaystyle\frac{19840}{81}\zeta_{3},$ (A.211) $\displaystyle
c_{v,2,1}(0,-1)$ $\displaystyle=$
$\displaystyle-\frac{936028}{243}-\frac{379792}{243}\zeta_{2}-\frac{300856}{405}\zeta_{2}^{2}+\frac{128}{9}\zeta_{2}^{2}l_{2}+\frac{10952}{9}\zeta_{5}$
(A.212)
$\displaystyle+\frac{11200}{81}\zeta_{2}l_{2}^{2}-\frac{5600}{243}l_{2}^{4}-\frac{44800}{81}a_{4}-\frac{2901776}{729}\zeta_{3}$
$\displaystyle+\frac{1908592}{243}\frac{\zeta_{3}}{\pi^{2}}-\frac{179024}{405}\zeta_{2}\zeta_{3}-\frac{38752}{81}\zeta_{3}l_{2}-\frac{85568}{81}\frac{\zeta_{3}^{2}}{\pi^{2}},$
$\displaystyle c_{v,2,1}(0,0)$ $\displaystyle=$
$\displaystyle\frac{102440}{27}-\frac{1816256}{243}\zeta_{2}-\frac{47008}{405}\zeta_{2}^{2}-\frac{56288}{105}\zeta_{2}^{3}+\frac{256}{9}\zeta_{2}^{2}l_{2}$
(A.213)
$\displaystyle+\frac{10112}{81}\zeta_{2}l_{2}^{2}-512\zeta_{2}^{2}l_{2}^{2}-\frac{5056}{243}l_{2}^{4}+\frac{256}{3}\zeta_{2}l_{2}^{4}$
$\displaystyle-\frac{40448}{81}a_{4}+2048a_{4}\zeta_{2}-\frac{6183424}{729}\zeta_{3}+\frac{2291072}{243}\frac{\zeta_{3}}{\pi^{2}}$
$\displaystyle-\frac{627088}{135}\zeta_{2}\zeta_{3}-\frac{72128}{81}\zeta_{3}l_{2}+1792\zeta_{2}\zeta_{3}l_{2}+\frac{2048}{3}\zeta_{3}^{2}$
$\displaystyle-\frac{14336}{81}\frac{\zeta_{3}^{2}}{\pi^{2}}+\frac{375584}{27}\zeta_{5}-10752\frac{\zeta_{3}\zeta_{5}}{\pi^{2}},$
$\displaystyle c_{v,2,1}(0,1)$ $\displaystyle=$
$\displaystyle\frac{19646248}{2187}-\frac{826384}{81}\zeta_{2}-\frac{6101096}{405}\zeta_{2}^{2}-\frac{102272}{105}\zeta_{2}^{3}$
(A.214)
$\displaystyle+\frac{1408}{9}\zeta_{2}^{2}l_{2}+\frac{32192}{9}\zeta_{2}l_{2}^{2}-\frac{16096}{27}l_{2}^{4}+6944\frac{\zeta_{3}\zeta_{5}}{\pi^{2}}$
$\displaystyle-\frac{128768}{9}a_{4}-\frac{6590224}{729}\zeta_{3}-\frac{1120700}{243}\frac{\zeta_{3}}{\pi^{2}}-\frac{786136}{27}\zeta_{5}$
$\displaystyle+\frac{3236608}{405}\zeta_{2}\zeta_{3}-7392\zeta_{3}l_{2}-\frac{640}{3}\zeta_{3}^{2}+\frac{2202368}{81}\frac{\zeta_{3}^{2}}{\pi^{2}},$
$\displaystyle c_{v,2,1}(0,2)$ $\displaystyle=$
$\displaystyle\frac{777783418}{6561}+\frac{65024a_{4}}{9}+\frac{8128l_{2}^{4}}{27}-\frac{1764008\zeta_{2}}{243}+4096a_{4}\zeta_{2}-\frac{16256}{9}l_{2}^{2}\zeta_{2}$
(A.215)
$\displaystyle+\frac{512l_{2}^{4}\zeta_{2}}{3}+\frac{163568\zeta_{2}^{2}}{27}+\frac{512l_{2}\zeta_{2}^{2}}{9}-1024l_{2}^{2}\zeta_{2}^{2}-\frac{112576\zeta_{2}^{3}}{105}-\frac{29704720\zeta_{3}}{729}$
$\displaystyle+\frac{26432l_{2}\zeta_{3}}{9}+\frac{110330920\zeta_{3}}{2187\pi^{2}}-\frac{3101792}{405}\zeta_{2}\zeta_{3}+3584l_{2}\zeta_{2}\zeta_{3}+\frac{4096\zeta_{3}^{2}}{3}$
$\displaystyle-\frac{674240\zeta_{3}^{2}}{81\pi^{2}}+\frac{762688\zeta_{5}}{27}-\frac{21504\zeta_{3}\zeta_{5}}{\pi^{2}}$
$\displaystyle c_{v,2,1}(0,3)$ $\displaystyle=$
$\displaystyle\frac{8943651091}{729000}-\frac{481280a_{4}}{81}-\frac{60160l_{2}^{4}}{243}-\frac{8824004\zeta_{2}}{729}+\frac{120320l_{2}^{2}\zeta_{2}}{81}$
(A.216)
$\displaystyle-\frac{3384272\zeta_{2}^{2}}{135}+\frac{512l_{2}\zeta_{2}^{2}}{3}-\frac{102272\zeta_{2}^{3}}{105}-\frac{9461092\zeta_{3}}{729}-\frac{363776l_{2}\zeta_{3}}{81}$
$\displaystyle-\frac{102560675\zeta_{3}}{4374\pi^{2}}+\frac{2475008\zeta_{2}\zeta_{3}}{405}-\frac{640\zeta_{3}^{2}}{3}-\frac{151648\zeta_{3}^{2}}{81\pi^{2}}-\frac{479104\zeta_{5}}{27}$
$\displaystyle+\frac{6944\zeta_{3}\zeta_{5}}{\pi^{2}}.$

##  Appendix B Expansions at $x=0$ ($\hat{s}\rightarrow-\infty$)

In Eq. (3.48), we showed the expansion up to $O(x^{3})$ of the non-solvable
parts of $F^{(0)}_{V,1}(x)$. In this appendix, we show the expansions up to
$O(x^{3})$ of the remaining form factors,

$\displaystyle F^{(0)}_{V,2}(x)$ $\displaystyle=$ $\displaystyle F^{(0),\rm
sol}_{V,2}(x)+n_{h}\biggl{\\{}-\frac{81920}{81}l_{2}\zeta_{2}+\biggl{[}-3072l_{2}\zeta_{2}^{2}-64\zeta_{2}\zeta_{3}-672\zeta_{5}$
(B.1)
$\displaystyle+\frac{2176}{81}l_{2}^{4}+\frac{9248}{45}\zeta_{2}^{2}-\frac{10112}{27}l_{2}^{2}\zeta_{2}+\frac{17408}{27}a_{4}-\frac{18016}{9}\zeta_{3}$
$\displaystyle+\frac{40768}{27}l_{2}\zeta_{2}+\frac{1091648}{243}+\frac{1837100}{243}\zeta_{2}+\biggl{(}\frac{2048}{9}l_{2}\zeta_{2}+\frac{7936}{9}\zeta_{3}$
$\displaystyle+\frac{14368}{9}\zeta_{2}-\frac{257264}{243}\biggr{)}\ln(x)+\biggl{(}-\frac{4456}{9}+\frac{5120}{27}\zeta_{2}\biggr{)}\ln^{2}(x)$
$\displaystyle-\frac{4672}{27}\ln^{3}(x)-\frac{880}{81}\ln^{4}(x)\biggr{]}x+\biggl{[}-4096a_{4}\zeta_{2}+1024l_{2}^{2}\zeta_{2}^{2}$
$\displaystyle+256\zeta_{3}^{2}-\frac{512}{3}l_{2}^{4}\zeta_{2}-\frac{19456}{243}l_{2}^{4}+\frac{23872}{105}\zeta_{2}^{3}+\frac{83968}{9}l_{2}\zeta_{2}^{2}$
$\displaystyle+\frac{113216}{27}\zeta_{2}\zeta_{3}+\frac{114944}{81}l_{2}^{2}\zeta_{2}-\frac{155648}{81}a_{4}-\frac{295936}{9}l_{2}\zeta_{2}$
$\displaystyle-\frac{1162336}{27}\zeta_{5}-\frac{2363776}{81}\zeta_{3}-\frac{2463584}{243}-\frac{7840040}{81}\zeta_{2}$
$\displaystyle+\frac{10280288}{405}\zeta_{2}^{2}+\biggl{(}-512\zeta_{5}-\frac{256}{9}l_{2}^{2}\zeta_{2}+\frac{896}{27}l_{2}^{4}+\frac{7168}{9}a_{4}$
$\displaystyle+\frac{29376}{5}\zeta_{2}^{2}-\frac{254848}{81}\zeta_{3}-\frac{257792}{27}l_{2}\zeta_{2}-\frac{817856}{81}\zeta_{2}$
$\displaystyle-\frac{1304960}{81}\biggr{)}\ln(x)+\biggl{(}-1024a_{4}+256l_{2}^{2}\zeta_{2}-\frac{128}{3}l_{2}^{4}+\frac{992}{5}\zeta_{2}^{2}$
$\displaystyle-\frac{34304}{9}l_{2}\zeta_{2}+\frac{48896}{27}\zeta_{3}-\frac{71200}{27}\zeta_{2}-\frac{684640}{243}\biggr{)}\ln^{2}(x)$
$\displaystyle+\biggl{(}-256l_{2}\zeta_{2}-\frac{64}{3}\zeta_{3}-\frac{64384}{81}\zeta_{2}+\frac{350528}{243}\biggr{)}\ln^{3}(x)+\frac{22976}{243}\ln^{4}(x)$
$\displaystyle-\frac{176}{405}\ln^{5}(x)\biggr{]}x^{2}+\biggl{[}20480a_{4}\zeta_{2}-5120l_{2}^{2}\zeta_{2}^{2}+\frac{2560}{3}l_{2}^{4}\zeta_{2}$
$\displaystyle-\frac{207872}{9}l_{2}\zeta_{2}^{2}+\frac{355520}{3}l_{2}\zeta_{2}+\frac{364096}{105}\zeta_{2}^{3}+\frac{548480}{243}l_{2}^{4}+\frac{766304}{3}\zeta_{5}$
$\displaystyle-\frac{647488}{27}\zeta_{2}\zeta_{3}-\frac{1356160}{81}l_{2}^{2}\zeta_{2}+\frac{4387840}{81}a_{4}+\frac{10194944}{81}\zeta_{3}$
$\displaystyle-\frac{12707392}{81}\zeta_{2}^{2}+\frac{92640452}{243}\zeta_{2}+\frac{204738082}{6561}+\biggl{(}-768\zeta_{2}\zeta_{3}$
$\displaystyle+10112\zeta_{5}-\frac{1792}{27}l_{2}^{4}-\frac{13312}{9}l_{2}^{2}\zeta_{2}-\frac{14336}{9}a_{4}+\frac{114944}{27}l_{2}\zeta_{2}$
$\displaystyle-\frac{5501312}{135}\zeta_{2}^{2}+\frac{5769280}{81}\zeta_{3}+\frac{10840352}{243}\zeta_{2}+\frac{25002020}{2187}\biggr{)}\ln(x)$
$\displaystyle+\biggl{(}5120a_{4}-1280l_{2}^{2}\zeta_{2}+\frac{640}{3}l_{2}^{4}-\frac{6496}{5}\zeta_{2}^{2}+\frac{160000}{9}l_{2}\zeta_{2}$
$\displaystyle-\frac{279616}{81}\zeta_{2}-\frac{351296}{27}\zeta_{3}+\frac{14492732}{729}\biggr{)}\ln^{2}(x)$
$\displaystyle+\biggl{(}2816l_{2}\zeta_{2}+\frac{640}{3}\zeta_{3}+\frac{152384}{27}\zeta_{2}-\frac{6464984}{729}\biggr{)}\ln^{3}(x)$
$\displaystyle+\biggl{(}16\zeta_{2}-\frac{342692}{243}\biggr{)}\ln^{4}(x)+\frac{6872}{81}\ln^{5}(x)+\frac{68}{45}\ln^{6}(x)\biggr{]}x^{3}+O(x^{4})\biggr{\\}},$
$\displaystyle F^{(0)}_{A,1}(x)$ $\displaystyle=$ $\displaystyle F^{(0),\rm
sol}_{A,1}(x)+n_{h}\biggl{\\{}-\frac{44288}{81x}l_{2}\zeta_{2}-1024l_{2}\zeta_{2}^{2}-\frac{1120}{81}l_{2}^{2}\zeta_{2}-\frac{1168}{243}l_{2}^{4}$
(B.2)
$\displaystyle-\frac{1640}{9}\zeta_{2}\zeta_{3}+\frac{3964}{9}\zeta_{5}+\frac{5068}{27}\zeta_{2}^{2}-\frac{7226}{9}-\frac{9344}{81}a_{4}+\frac{46324}{81}\zeta_{3}$
$\displaystyle+\frac{255952}{81}l_{2}\zeta_{2}+\frac{1862569}{243}\zeta_{2}+\biggl{(}\frac{64}{27}l_{2}^{4}+\frac{512}{9}a_{4}-\frac{512}{9}l_{2}^{2}\zeta_{2}$
$\displaystyle+\frac{15520}{27}l_{2}\zeta_{2}+\frac{21728}{135}\zeta_{2}^{2}+\frac{34660}{243}\zeta_{2}-\frac{55676}{81}\zeta_{3}+\frac{670460}{243}\biggr{)}\ln(x)$
$\displaystyle+\biggl{(}\frac{128}{9}l_{2}\zeta_{2}-\frac{512}{9}\zeta_{3}+\frac{2264}{81}\zeta_{2}+\frac{18274}{27}\biggr{)}\ln^{2}(x)$
$\displaystyle+\biggl{(}60-\frac{32}{3}\zeta_{2}\biggr{)}\ln^{3}(x)-\frac{172}{243}\ln^{4}(x)-\frac{176}{81}\ln^{5}(x)$
$\displaystyle+\biggl{[}\frac{1600}{243}l_{2}^{4}+\frac{12352}{81}l_{2}^{2}\zeta_{2}+\frac{12800}{81}a_{4}+\frac{16000}{27}\zeta_{2}\zeta_{3}+\frac{27392}{9}l_{2}\zeta_{2}^{2}$
$\displaystyle-\frac{132320}{27}\zeta_{5}+\frac{194336}{81}\zeta_{2}^{2}-\frac{442960}{81}\zeta_{3}-\frac{850912}{81}l_{2}\zeta_{2}+\frac{1637576}{243}$
$\displaystyle-\frac{2210888}{81}\zeta_{2}+\biggl{(}\frac{640}{27}l_{2}^{4}-\frac{1664}{9}l_{2}^{2}\zeta_{2}+\frac{2912}{5}\zeta_{2}^{2}+\frac{5120}{9}a_{4}$
$\displaystyle-\frac{13664}{81}\zeta_{3}-\frac{53248}{27}l_{2}\zeta_{2}-\frac{167312}{81}\zeta_{2}-\frac{1355456}{243}\biggr{)}\ln(x)$
$\displaystyle+\biggl{(}-\frac{1904}{9}\zeta_{2}-\frac{3520}{9}l_{2}\zeta_{2}+\frac{4432}{27}\zeta_{3}-\frac{86036}{81}\biggr{)}\ln^{2}(x)$
$\displaystyle+\biggl{(}-\frac{7664}{81}\zeta_{2}-\frac{13088}{81}\biggr{)}\ln^{3}(x)+\frac{17200}{243}\ln^{4}(x)+\frac{44}{405}\ln^{5}(x)\biggr{]}x$
$\displaystyle+\biggl{[}1024a_{4}\zeta_{2}+26304l_{2}\zeta_{2}-256l_{2}^{2}\zeta_{2}^{2}-48\zeta_{3}^{2}+\frac{128}{3}l_{2}^{4}\zeta_{2}-\frac{24320}{3}l_{2}\zeta_{2}^{2}$
$\displaystyle+\frac{32176}{105}\zeta_{2}^{3}+\frac{36608}{243}l_{2}^{4}-\frac{87664}{27}\zeta_{2}\zeta_{3}-\frac{100864}{81}l_{2}^{2}\zeta_{2}-\frac{129970051}{13122}$
$\displaystyle+\frac{292864}{81}a_{4}+\frac{1359904}{81}\zeta_{3}+\frac{4874924}{81}\zeta_{2}-\frac{5137136}{405}\zeta_{2}^{2}+\frac{249592}{9}\zeta_{5}$
$\displaystyle+\biggl{(}6320\zeta_{2}-128\zeta_{2}\zeta_{3}+576\zeta_{5}-\frac{1280}{27}l_{2}^{4}+\frac{3328}{9}l_{2}^{2}\zeta_{2}-\frac{10240}{9}a_{4}$
$\displaystyle-\frac{36608}{27}l_{2}\zeta_{2}-\frac{138448}{45}\zeta_{2}^{2}+\frac{520232}{81}\zeta_{3}+\frac{22789421}{2187}\biggr{)}\ln(x)$
$\displaystyle+\biggl{(}256a_{4}-64l_{2}^{2}\zeta_{2}-104\zeta_{2}^{2}+\frac{32}{3}l_{2}^{4}+\frac{12032}{9}l_{2}\zeta_{2}-\frac{31048}{27}\zeta_{2}$
$\displaystyle-\frac{43904}{27}\zeta_{3}+\frac{4494101}{729}\biggr{)}\ln^{2}(x)+\biggl{(}192l_{2}\zeta_{2}+\frac{32}{3}\zeta_{3}+\frac{43696}{81}\zeta_{2}$
$\displaystyle+\frac{91018}{729}\biggr{)}\ln^{3}(x)+\biggl{(}4\zeta_{2}-\frac{52199}{243}\biggr{)}\ln^{4}(x)+\frac{242}{405}\ln^{5}(x)\biggr{]}x^{2}$
$\displaystyle+\biggl{[}-2048a_{4}\zeta_{2}+512l_{2}^{2}\zeta_{2}^{2}-512\zeta_{3}^{2}-\frac{256}{3}l_{2}^{4}\zeta_{2}-\frac{18656}{15}\zeta_{2}^{3}$
$\displaystyle-\frac{125888}{243}l_{2}^{4}+\frac{136960}{9}l_{2}\zeta_{2}^{2}+\frac{162944}{27}\zeta_{2}\zeta_{3}+\frac{298432}{81}l_{2}^{2}\zeta_{2}$
$\displaystyle-\frac{468128}{9}l_{2}\zeta_{2}-\frac{1007104}{81}a_{4}-\frac{1501472}{27}\zeta_{5}-\frac{2243680}{81}\zeta_{3}$
$\displaystyle+\frac{2364736}{81}\zeta_{2}^{2}-\frac{47478772}{6561}-\frac{80144464}{729}\zeta_{2}+\biggl{(}-640\zeta_{2}\zeta_{3}$
$\displaystyle-1728\zeta_{5}+\frac{5632}{27}l_{2}^{4}-\frac{13952}{9}l_{2}^{2}\zeta_{2}+\frac{45056}{9}a_{4}+\frac{206204}{243}$
$\displaystyle+\frac{213248}{27}l_{2}\zeta_{2}+\frac{1104608}{135}\zeta_{2}^{2}-\frac{2051936}{81}\zeta_{3}-\frac{5104624}{243}\zeta_{2}\biggr{)}\ln(x)$
$\displaystyle+\biggl{(}-512a_{4}+128l_{2}^{2}\zeta_{2}-\frac{64}{3}l_{2}^{4}+\frac{176}{5}\zeta_{2}^{2}-\frac{21824}{9}l_{2}\zeta_{2}+\frac{75472}{27}\zeta_{3}$
$\displaystyle+\frac{493280}{81}\zeta_{2}-\frac{9056684}{729}\biggr{)}\ln^{2}(x)+\biggl{(}\frac{64}{3}\zeta_{3}-384l_{2}\zeta_{2}-\frac{35312}{81}$
$\displaystyle-\frac{111536}{81}\zeta_{2}\biggr{)}\ln^{3}(x)+\biggl{(}\frac{5900}{9}-8\zeta_{2}\biggr{)}\ln^{4}(x)-\frac{3712}{405}\ln^{5}(x)$
$\displaystyle-\frac{34}{45}\ln^{6}(x)\biggr{]}x^{3}+O(x^{4})\biggr{\\}},$
$\displaystyle F^{(0)}_{A,2}(x)$ $\displaystyle=$ $\displaystyle F^{(0),\rm
sol}_{A,2}(x)+n_{h}\biggl{\\{}\frac{13312}{27}l_{2}\zeta_{2}+\biggl{[}-\frac{128}{3}l_{2}^{2}\zeta_{2}-\frac{128}{9}l_{2}^{4}-\frac{22528}{135}\zeta_{2}^{2}$
(B.3)
$\displaystyle-\frac{1024}{3}a_{4}+\frac{96032}{27}\zeta_{3}+\frac{97984}{27}l_{2}\zeta_{2}+\frac{273376}{81}+\frac{1417540}{81}\zeta_{2}$
$\displaystyle+\biggl{(}\frac{2816}{9}l_{2}\zeta_{2}-\frac{5888}{27}\zeta_{3}+\frac{18016}{27}\zeta_{2}+\frac{1883216}{243}\biggr{)}\ln(x)$
$\displaystyle+\biggl{(}\frac{992}{27}\zeta_{2}+\frac{347672}{243}\biggr{)}\ln^{2}(x)-\frac{2368}{81}\ln^{3}(x)-\frac{1360}{27}\ln^{4}(x)\biggr{]}x$
$\displaystyle+\biggl{[}-\frac{5888}{81}l_{2}^{2}\zeta_{2}+\frac{13312}{243}l_{2}^{4}+\frac{14144}{27}\zeta_{2}\zeta_{3}+\frac{81920}{9}l_{2}\zeta_{2}^{2}+\frac{106496}{81}a_{4}$
$\displaystyle-\frac{397408}{27}\zeta_{5}-\frac{579328}{27}l_{2}\zeta_{2}-\frac{920960}{81}\zeta_{3}+\frac{1591712}{81}+\frac{3029984}{405}\zeta_{2}^{2}$
$\displaystyle-\frac{13386440}{243}\zeta_{2}+\biggl{(}\frac{2048}{3}\zeta_{2}^{2}+\frac{3712}{27}l_{2}^{4}-\frac{8960}{9}l_{2}^{2}\zeta_{2}+\frac{29696}{9}a_{4}$
$\displaystyle-\frac{41216}{9}l_{2}\zeta_{2}+\frac{45824}{81}\zeta_{3}-\frac{49856}{81}-\frac{387008}{81}\zeta_{2}\biggr{)}\ln(x)$
$\displaystyle+\biggl{(}\frac{1280}{27}\zeta_{3}-\frac{14080}{9}l_{2}\zeta_{2}-\frac{79136}{81}\zeta_{2}-\frac{351616}{243}\biggr{)}\ln^{2}(x)$
$\displaystyle+\biggl{(}\frac{5248}{27}-\frac{24896}{81}\zeta_{2}\biggr{)}\ln^{3}(x)+\frac{22624}{243}\ln^{4}(x)+\frac{176}{405}\ln^{5}(x)\biggr{]}x^{2}$
$\displaystyle+\biggl{[}4096a_{4}\zeta_{2}+53824l_{2}\zeta_{2}-1024l_{2}^{2}\zeta_{2}^{2}-13696\zeta_{2}\zeta_{3}-1408\zeta_{3}^{2}$
$\displaystyle+\frac{512}{3}l_{2}^{4}\zeta_{2}-\frac{3776}{105}\zeta_{2}^{3}-\frac{53248}{9}l_{2}\zeta_{2}^{2}+\frac{108416}{243}l_{2}^{4}-\frac{206464}{81}l_{2}^{2}\zeta_{2}$
$\displaystyle+\frac{867328}{81}a_{4}+\frac{1905088}{27}\zeta_{5}+\frac{4172704}{81}\zeta_{3}-\frac{5531966}{2187}+\frac{7887212}{81}\zeta_{2}$
$\displaystyle-\frac{12842176}{405}\zeta_{2}^{2}+\biggl{(}-2304\zeta_{2}\zeta_{3}+1152\zeta_{5}-\frac{1024}{3}l_{2}^{2}\zeta_{2}+\frac{1280}{9}l_{2}^{4}$
$\displaystyle+\frac{10240}{3}a_{4}-\frac{345856}{27}l_{2}\zeta_{2}-\frac{2236736}{135}\zeta_{2}^{2}+\frac{2694272}{81}\zeta_{3}+\frac{4301152}{243}\zeta_{2}$
$\displaystyle+\frac{24011348}{729}\biggr{)}\ln(x)+\biggl{(}1024a_{4}-256l_{2}^{2}\zeta_{2}+\frac{128}{3}l_{2}^{4}-\frac{3808}{5}\zeta_{2}^{2}$
$\displaystyle+\frac{15872}{9}l_{2}\zeta_{2}-\frac{48256}{9}\zeta_{3}-\frac{100384}{27}\zeta_{2}+\frac{126532}{9}\biggr{)}\ln^{2}(x)$
$\displaystyle+\biggl{(}768l_{2}\zeta_{2}+128\zeta_{3}-\frac{76184}{243}+\frac{144704}{81}\zeta_{2}\biggr{)}\ln^{3}(x)$
$\displaystyle+\biggl{(}16\zeta_{2}-\frac{66932}{81}\biggr{)}\ln^{4}(x)+\frac{184}{15}\ln^{5}(x)-\frac{68}{45}\ln^{6}(x)\biggr{]}x^{3}+O(x^{4})\biggr{\\}},$
$\displaystyle F^{(0)}_{S}(x)$ $\displaystyle=$ $\displaystyle F^{(0),\rm
sol}_{S}(x)+n_{h}\biggl{\\{}-\frac{34304}{81x}l_{2}\zeta_{2}-1024l_{2}\zeta_{2}^{2}-\frac{256}{81}l_{2}^{2}\zeta_{2}-\frac{304}{243}l_{2}^{4}$
(B.4)
$\displaystyle-\frac{1640}{9}\zeta_{2}\zeta_{3}-\frac{2432}{81}a_{4}+\frac{3964}{9}\zeta_{5}+\frac{10324}{45}\zeta_{2}^{2}+\frac{107668}{81}\zeta_{3}$
$\displaystyle+\frac{209600}{81}l_{2}\zeta_{2}-\frac{1676170}{243}+\frac{1828108}{243}\zeta_{2}+\biggl{(}\frac{64}{27}l_{2}^{4}+\frac{512}{9}a_{4}$
$\displaystyle-\frac{512}{9}l_{2}^{2}\zeta_{2}-\frac{5876}{243}\zeta_{2}+\frac{13408}{27}l_{2}\zeta_{2}+\frac{21728}{135}\zeta_{2}^{2}-\frac{51260}{81}\zeta_{3}$
$\displaystyle+\frac{22184}{27}\biggr{)}\ln(x)+\biggl{(}\frac{128}{9}l_{2}\zeta_{2}-\frac{512}{9}\zeta_{3}+\frac{1520}{81}\zeta_{2}+\frac{77548}{243}\biggr{)}\ln^{2}(x)$
$\displaystyle+\biggl{(}-\frac{32}{3}\zeta_{2}+\frac{5452}{81}\biggr{)}\ln^{3}(x)+\frac{2888}{243}\ln^{4}(x)-\frac{176}{81}\ln^{5}(x)$
$\displaystyle+\biggl{[}2304l_{2}\zeta_{2}^{2}-\frac{1472}{81}l_{2}^{4}+\frac{8128}{27}l_{2}^{2}\zeta_{2}-\frac{11776}{27}a_{4}+\frac{13328}{27}\zeta_{2}\zeta_{3}$
$\displaystyle-\frac{23896}{27}\zeta_{5}+\frac{62168}{405}\zeta_{2}^{2}-\frac{99776}{81}\zeta_{3}-\frac{534016}{81}l_{2}\zeta_{2}+\frac{1087216}{243}$
$\displaystyle-\frac{4558496}{243}\zeta_{2}+\biggl{(}64l_{2}^{2}\zeta_{2}-256a_{4}-\frac{32}{3}l_{2}^{4}+\frac{6176}{15}\zeta_{2}^{2}-\frac{40912}{27}\zeta_{2}$
$\displaystyle-\frac{25408}{27}l_{2}\zeta_{2}-\frac{44704}{81}\zeta_{3}-\frac{2297624}{243}\biggr{)}\ln(x)-\biggl{(}\frac{2344}{81}\zeta_{2}+\frac{138080}{81}$
$\displaystyle-\frac{4112}{27}\zeta_{3}\biggr{)}\ln^{2}(x)-\biggl{(}\frac{160}{9}\zeta_{2}+\frac{9088}{27}\biggr{)}\ln^{3}(x)+\frac{5456}{81}\ln^{4}(x)\biggr{]}x$
$\displaystyle+\biggl{[}-272\zeta_{3}^{2}-\frac{2144}{35}\zeta_{2}^{3}-\frac{24320}{3}l_{2}\zeta_{2}^{2}+\frac{34816}{243}l_{2}^{4}-\frac{41776}{27}\zeta_{2}\zeta_{3}$
$\displaystyle-\frac{100736}{81}l_{2}^{2}\zeta_{2}+\frac{259208}{27}\zeta_{5}+\frac{278528}{81}a_{4}+\frac{510944}{27}l_{2}\zeta_{2}+\frac{1004896}{81}\zeta_{3}$
$\displaystyle-\frac{1411504}{405}\zeta_{2}^{2}+\frac{11143964}{243}\zeta_{2}-\frac{62619035}{6561}+\biggl{(}-192\zeta_{2}\zeta_{3}-544\zeta_{5}$
$\displaystyle+\frac{512}{27}l_{2}^{4}-\frac{640}{9}l_{2}^{2}\zeta_{2}+\frac{4096}{9}a_{4}+\frac{12032}{9}l_{2}\zeta_{2}-\frac{19904}{27}\zeta_{2}^{2}+\frac{68696}{27}\zeta_{3}$
$\displaystyle+\frac{963968}{243}\zeta_{2}+\frac{15324602}{2187}\biggr{)}\ln(x)+\biggl{(}\frac{4736}{9}l_{2}\zeta_{2}-\frac{432}{5}\zeta_{2}^{2}-\frac{20480}{27}\zeta_{3}$
$\displaystyle-\frac{29176}{81}\zeta_{2}+\frac{4544258}{729}\biggr{)}\ln^{2}(x)-\biggl{(}\frac{32}{3}\zeta_{3}+\frac{2324}{729}-\frac{4544}{27}\zeta_{2}\biggr{)}\ln^{3}(x)$
$\displaystyle+\biggl{(}8\zeta_{2}-\frac{9028}{243}\biggr{)}\ln^{4}(x)-\frac{3032}{405}\ln^{5}(x)-\frac{17}{45}\ln^{6}(x)\biggr{]}x^{2}$
$\displaystyle+\biggl{[}544\zeta_{3}^{2}+\frac{4288}{35}\zeta_{2}^{3}+\frac{43264}{3}l_{2}\zeta_{2}^{2}+\frac{82448}{27}\zeta_{2}\zeta_{3}-\frac{2633152}{81}\zeta_{3}$
$\displaystyle+\frac{218560}{81}l_{2}^{2}\zeta_{2}-\frac{424736}{9}l_{2}\zeta_{2}-\frac{431096}{27}\zeta_{5}+\frac{652576}{81}\zeta_{2}^{2}-\frac{687616}{81}a_{4}$
$\displaystyle-\frac{85952}{243}l_{2}^{4}+\frac{29263327}{2187}-\frac{72816460}{729}\zeta_{2}+\biggl{(}384\zeta_{2}\zeta_{3}+1088\zeta_{5}$
$\displaystyle-\frac{64}{3}l_{2}^{2}\zeta_{2}-\frac{352}{9}l_{2}^{4}-\frac{2176}{3}l_{2}\zeta_{2}-\frac{2816}{3}a_{4}+\frac{530096}{135}\zeta_{2}^{2}-\frac{873344}{81}\zeta_{3}$
$\displaystyle-\frac{1344104}{81}\zeta_{2}-\frac{50224357}{2187}\biggr{)}\ln(x)+\biggl{(}\frac{864}{5}\zeta_{2}^{2}-\frac{3008}{3}l_{2}\zeta_{2}+\frac{9584}{9}\zeta_{3}$
$\displaystyle+\frac{142976}{81}\zeta_{2}-\frac{2236508}{243}\biggr{)}\ln^{2}(x)+\biggl{(}\frac{64}{3}\zeta_{3}-\frac{41008}{81}\zeta_{2}$
$\displaystyle-\frac{1157750}{729}\biggr{)}\ln^{3}(x)-\biggl{(}16\zeta_{2}-\frac{115906}{243}\biggr{)}\ln^{4}(x)+\frac{722}{135}\ln^{5}(x)$
$\displaystyle+\frac{34}{45}\ln^{6}(x)\biggr{]}x^{3}+O(x^{4})\biggr{\\}},$
$\displaystyle F^{(0)}_{P}(x)$ $\displaystyle=$ $\displaystyle F^{(0),\rm
sol}_{P}(x)+n_{h}\biggl{\\{}-\frac{54272}{81x}l_{2}\zeta_{2}-1024l_{2}\zeta_{2}^{2}-\frac{256}{81}l_{2}^{2}\zeta_{2}-\frac{304}{243}l_{2}^{4}$
(B.5)
$\displaystyle-\frac{1640}{9}\zeta_{2}\zeta_{3}-\frac{2432}{81}a_{4}+\frac{3964}{9}\zeta_{5}+\frac{10324}{45}\zeta_{2}^{2}+\frac{202432}{81}l_{2}\zeta_{2}$
$\displaystyle+\frac{107668}{81}\zeta_{3}-\frac{1676170}{243}+\frac{1828108}{243}\zeta_{2}+\biggl{(}\frac{64}{27}l_{2}^{4}+\frac{512}{9}a_{4}-\frac{512}{9}l_{2}^{2}\zeta_{2}$
$\displaystyle-\frac{5876}{243}\zeta_{2}+\frac{13408}{27}l_{2}\zeta_{2}+\frac{21728}{135}\zeta_{2}^{2}+\frac{22184}{27}-\frac{51260}{81}\zeta_{3}\biggr{)}\ln(x)$
$\displaystyle+\biggl{(}\frac{128}{9}l_{2}\zeta_{2}-\frac{512}{9}\zeta_{3}+\frac{1520}{81}\zeta_{2}+\frac{77548}{243}\biggr{)}\ln^{2}(x)$
$\displaystyle+\biggl{(}-\frac{32}{3}\zeta_{2}+\frac{5452}{81}\biggr{)}\ln^{3}(x)+\frac{2888}{243}\ln^{4}(x)-\frac{176}{81}\ln^{5}(x)$
$\displaystyle+\biggl{[}768l_{2}\zeta_{2}^{2}-\frac{128}{9}l_{2}^{4}+\frac{448}{3}l_{2}^{2}\zeta_{2}-\frac{1024}{3}a_{4}+\frac{12464}{27}\zeta_{2}\zeta_{3}-\frac{32968}{27}\zeta_{5}$
$\displaystyle-\frac{68672}{81}\zeta_{3}+\frac{180392}{405}\zeta_{2}^{2}-\frac{279424}{81}l_{2}\zeta_{2}+\frac{853856}{243}-\frac{1159744}{243}\zeta_{2}$
$\displaystyle+\biggl{(}-256a_{4}+64l_{2}^{2}\zeta_{2}-\frac{32}{3}l_{2}^{4}+\frac{6176}{15}\zeta_{2}^{2}-\frac{14512}{27}\zeta_{2}-\frac{18112}{27}l_{2}\zeta_{2}$
$\displaystyle-\frac{33952}{81}\zeta_{3}-\frac{376456}{243}\biggr{)}\ln(x)+\biggl{(}\frac{3632}{243}+\frac{4112}{27}\zeta_{3}+\frac{4136}{81}\zeta_{2}\biggr{)}\ln^{2}(x)$
$\displaystyle+\biggl{(}-\frac{160}{9}\zeta_{2}-\frac{18208}{81}\biggr{)}\ln^{3}(x)+\frac{1808}{81}\ln^{4}(x)\biggr{]}x$
$\displaystyle+\biggl{[}304\zeta_{3}^{2}+\frac{2208}{7}\zeta_{2}^{3}+\frac{3952}{9}\zeta_{2}\zeta_{3}+\frac{8200}{3}\zeta_{5}+\frac{10912}{9}l_{2}\zeta_{2}+\frac{17024}{243}l_{2}^{4}$
$\displaystyle-\frac{18688}{9}l_{2}\zeta_{2}^{2}-\frac{43856}{45}\zeta_{2}^{2}-\frac{51328}{81}l_{2}^{2}\zeta_{2}+\frac{136192}{81}a_{4}-\frac{215776}{81}\zeta_{3}$
$\displaystyle+\frac{952988}{243}\zeta_{2}-\frac{1907579}{6561}+\biggl{(}448\zeta_{2}\zeta_{3}+288\zeta_{5}-\frac{128}{3}l_{2}^{2}\zeta_{2}-\frac{128}{9}l_{2}^{4}$
$\displaystyle-\frac{1024}{3}a_{4}-\frac{14080}{27}l_{2}\zeta_{2}+\frac{37984}{27}\zeta_{2}^{2}-\frac{126008}{81}\zeta_{3}-\frac{129382}{2187}$
$\displaystyle-\frac{160576}{243}\zeta_{2}\biggr{)}\ln(x)+\biggl{(}\frac{432}{5}\zeta_{2}^{2}+\frac{1024}{9}l_{2}\zeta_{2}-\frac{7072}{27}\zeta_{3}-\frac{58168}{81}\zeta_{2}$
$\displaystyle+\frac{1143650}{729}\biggr{)}\ln^{2}(x)+\biggl{(}-\frac{64}{3}\zeta_{3}-\frac{4928}{81}\zeta_{2}+\frac{224332}{729}\biggr{)}\ln^{3}(x)$
$\displaystyle+\frac{4124}{81}\ln^{4}(x)-\frac{304}{135}\ln^{5}(x)+\frac{17}{45}\ln^{6}(x)\biggr{]}x^{2}$
$\displaystyle+\biggl{[}608\zeta_{3}^{2}+\frac{4416}{7}\zeta_{2}^{3}+\frac{6400}{9}l_{2}\zeta_{2}^{2}-\frac{6560}{3}l_{2}\zeta_{2}+\frac{8320}{81}l_{2}^{4}-\frac{11456}{27}l_{2}^{2}\zeta_{2}$
$\displaystyle-\frac{19048}{27}\zeta_{5}+\frac{33328}{27}\zeta_{2}\zeta_{3}+\frac{66560}{27}a_{4}-\frac{120800}{81}\zeta_{2}^{2}+\frac{275728}{81}\zeta_{3}$
$\displaystyle+\frac{7061252}{729}\zeta_{2}+\frac{57246137}{6561}+\biggl{(}896\zeta_{2}\zeta_{3}+576\zeta_{5}-\frac{1312}{27}l_{2}^{4}$
$\displaystyle+\frac{1856}{9}l_{2}^{2}\zeta_{2}-\frac{10496}{9}a_{4}-\frac{25472}{9}l_{2}\zeta_{2}+\frac{113296}{135}\zeta_{2}^{2}-\frac{187424}{81}\zeta_{3}$
$\displaystyle+\frac{336520}{243}\zeta_{2}+\frac{529369}{729}\biggr{)}\ln(x)+\biggl{(}\frac{864}{5}\zeta_{2}^{2}-\frac{1328}{3}\zeta_{2}+\frac{1600}{9}l_{2}\zeta_{2}$
$\displaystyle+\frac{2384}{9}\zeta_{3}+\frac{579628}{729}\biggr{)}\ln^{2}(x)+\biggl{(}\frac{400}{9}\zeta_{2}-\frac{128}{3}\zeta_{3}-\frac{42146}{243}\biggr{)}\ln^{3}(x)$
$\displaystyle+\frac{15370}{243}\ln^{4}(x)+\frac{238}{405}\ln^{5}(x)+\frac{34}{45}\ln^{6}(x)\biggr{]}x^{3}+O(x^{4})\biggr{\\}}.$

As we did in the case of Eq. (3.49) for $F^{n_{h}}_{V,1}(x)$, we can also
perform the corresponding expansions in $\hat{s}$ up to $O(\hat{s}^{-3})$ for
the other form factors. These expansions include the solvable parts, as well
as the renormalization and infrared subtraction terms. One obtains

$\displaystyle F^{n_{h}}_{V,2}(\hat{s})$ $\displaystyle=$
$\displaystyle\biggl{[}832\zeta_{5}+\frac{256}{27}l_{2}^{4}-\frac{512}{9}l_{2}^{2}\zeta_{2}+\frac{2048}{9}a_{4}-\frac{32896}{27}\zeta_{2}^{2}+\frac{47776}{27}\zeta_{3}$
(B.6)
$\displaystyle+\frac{114688}{9}l_{2}\zeta_{2}-\frac{535360}{243}-\frac{2382464}{243}\zeta_{2}+\biggl{(}\frac{8192}{27}\zeta_{3}-\frac{54592}{81}\zeta_{2}$
$\displaystyle-\frac{176576}{243}\biggr{)}l_{s}+\biggl{(}-\frac{1696}{27}\zeta_{2}-\frac{1808}{243}\biggr{)}l_{s}^{2}+\frac{1664}{81}l_{s}^{3}+\frac{320}{81}l_{s}^{4}\biggr{]}\frac{1}{\hat{s}}$
$\displaystyle+\biggl{[}-\frac{35264}{105}\zeta_{2}^{3}-\frac{36352}{243}l_{2}^{4}+\frac{72704}{81}l_{2}^{2}\zeta_{2}+\frac{1091648}{81}\zeta_{3}-\frac{116192}{81}\zeta_{2}^{2}$
$\displaystyle-\frac{6016}{27}\zeta_{5}+\frac{7808}{9}\zeta_{2}\zeta_{3}-\frac{290816}{81}a_{4}-\frac{481984}{81}+\frac{2732032}{27}l_{2}\zeta_{2}$
$\displaystyle+256\zeta_{3}^{2}-\frac{23493952}{243}\zeta_{2}+\biggl{(}\frac{724352}{81}\zeta_{3}-\frac{508864}{81}\zeta_{2}-\frac{1299712}{81}$
$\displaystyle+\frac{768}{5}\zeta_{2}^{2}-512\zeta_{5}\biggr{)}l_{s}+\biggl{(}\frac{288}{5}\zeta_{2}^{2}+\frac{896}{3}\zeta_{3}-\frac{37792}{27}\zeta_{2}-\frac{389248}{243}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}-\frac{64}{3}\zeta_{3}-\frac{8320}{27}\zeta_{2}+\frac{159104}{243}\biggr{)}l_{s}^{3}+\frac{11648}{243}l_{s}^{4}-\frac{16}{81}l_{s}^{5}\biggr{]}\frac{1}{\hat{s}^{2}}$
$\displaystyle+\biggl{[}1024l_{2}\zeta_{2}^{2}+1024\zeta_{3}^{2}+\frac{65920}{9}\zeta_{2}\zeta_{3}-\frac{88768}{27}\zeta_{5}-\frac{202432}{35}\zeta_{2}^{3}$
$\displaystyle-\frac{226304}{81}l_{2}^{4}+\frac{452608}{27}l_{2}^{2}\zeta_{2}-\frac{1810432}{27}a_{4}+\frac{4540000}{81}\zeta_{3}$
$\displaystyle-\frac{5931872}{9}\zeta_{2}-\frac{7876978}{6561}+\frac{15570016}{405}\zeta_{2}^{2}+\frac{16488448}{27}l_{2}\zeta_{2}$
$\displaystyle+\biggl{(}2304\zeta_{2}\zeta_{3}-13440\zeta_{5}-\frac{1024}{27}l_{2}^{4}+\frac{2048}{9}l_{2}^{2}\zeta_{2}-\frac{8192}{9}a_{4}$
$\displaystyle+\frac{57472}{45}\zeta_{2}^{2}+\frac{212992}{27}l_{2}\zeta_{2}+\frac{6275776}{81}\zeta_{3}-\frac{10957184}{243}\zeta_{2}$
$\displaystyle-\frac{262638740}{2187}\biggr{)}l_{s}+\biggl{(}\frac{6432}{5}\zeta_{2}^{2}+\frac{104896}{27}\zeta_{3}-\frac{772160}{81}\zeta_{2}$
$\displaystyle-\frac{5223476}{729}\biggr{)}l_{s}^{2}+\biggl{(}-\frac{896}{3}\zeta_{3}-\frac{291136}{81}\zeta_{2}+\frac{5233208}{729}\biggr{)}l_{s}^{3}$
$\displaystyle+\biggl{(}-16\zeta_{2}+\frac{161668}{243}\biggr{)}l_{s}^{4}-\frac{16984}{405}l_{s}^{5}-\frac{68}{45}l_{s}^{6}\biggr{]}\frac{1}{\hat{s}^{3}}+O\left(\frac{1}{\hat{s}^{4}}\right),$
$\displaystyle F^{n_{h}}_{A,1}(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{64}{3}l_{2}\zeta_{2}^{2}-\frac{464}{3}\zeta_{2}\zeta_{3}-\frac{3808}{243}l_{2}^{4}-\frac{4856}{9}\zeta_{5}+\frac{5828}{27}\zeta_{2}^{2}+\frac{7616}{81}l_{2}^{2}\zeta_{2}$
(B.7)
$\displaystyle-\frac{30464}{81}a_{4}-\frac{52708}{81}\zeta_{3}-\frac{107008}{81}l_{2}\zeta_{2}+\frac{281176}{243}\zeta_{2}+\frac{461144}{243}$
$\displaystyle+\biggl{(}-\frac{64}{81}l_{2}^{4}+\frac{128}{27}l_{2}^{2}\zeta_{2}+\frac{368}{27}\zeta_{2}^{2}-\frac{512}{27}a_{4}-\frac{1804}{9}\zeta_{3}+\frac{47936}{243}\zeta_{2}$
$\displaystyle+\frac{171704}{243}\biggr{)}l_{s}+\biggl{(}-\frac{160}{9}\zeta_{3}+\frac{896}{81}\zeta_{2}+\frac{46064}{243}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}-\frac{416}{81}\zeta_{2}+\frac{4616}{243}\biggr{)}l_{s}^{3}-\frac{136}{81}l_{s}^{4}-\frac{32}{81}l_{s}^{5}+\biggl{[}-\frac{1600}{27}\zeta_{5}+\frac{3584}{243}l_{2}^{4}$
$\displaystyle-\frac{2240}{9}\zeta_{2}\zeta_{3}-\frac{7168}{81}l_{2}^{2}\zeta_{2}+\frac{14224}{81}\zeta_{2}^{2}+\frac{28672}{81}a_{4}-\frac{77576}{81}\zeta_{3}$
$\displaystyle-\frac{207760}{243}-\frac{625408}{27}l_{2}\zeta_{2}+\frac{5245840}{243}\zeta_{2}+\biggl{(}\frac{1696}{45}\zeta_{2}^{2}+\frac{86320}{81}\zeta_{2}$
$\displaystyle-\frac{118048}{81}\zeta_{3}+\frac{296048}{81}\biggr{)}l_{s}+\biggl{(}\frac{32}{9}\zeta_{2}+\frac{112}{9}\zeta_{3}+\frac{54928}{81}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}\frac{848}{27}\zeta_{2}+\frac{1088}{81}\biggr{)}l_{s}^{3}-\frac{5896}{243}l_{s}^{4}-\frac{4}{81}l_{s}^{5}\biggr{]}\frac{1}{\hat{s}}$
$\displaystyle+\biggl{[}-48\zeta_{3}^{2}-\frac{128}{3}l_{2}\zeta_{2}^{2}+\frac{16832}{81}l_{2}^{4}+\frac{30832}{105}\zeta_{2}^{3}-\frac{33664}{27}l_{2}^{2}\zeta_{2}$
$\displaystyle-\frac{36560}{27}\zeta_{2}\zeta_{3}-\frac{58952}{27}\zeta_{5}-\frac{70276}{81}\zeta_{3}+\frac{134656}{27}a_{4}-\frac{498592}{405}\zeta_{2}^{2}$
$\displaystyle-\frac{8287744}{81}l_{2}\zeta_{2}+\frac{23853932}{243}\zeta_{2}-\frac{78457723}{13122}$
$\displaystyle+\biggl{(}-288\zeta_{2}\zeta_{3}+656\zeta_{5}+\frac{128}{81}l_{2}^{4}-\frac{256}{27}l_{2}^{2}\zeta_{2}+\frac{1024}{27}a_{4}+\frac{7024}{45}\zeta_{2}^{2}$
$\displaystyle-\frac{19456}{27}l_{2}\zeta_{2}-\frac{658360}{81}\zeta_{3}+\frac{1181320}{243}\zeta_{2}+\frac{40251689}{2187}\biggr{)}l_{s}$
$\displaystyle+\biggl{(}-\frac{536}{5}\zeta_{2}^{2}+\frac{3272}{27}\zeta_{2}-\frac{9632}{27}\zeta_{3}+\frac{2847341}{729}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}\frac{32}{3}\zeta_{3}+\frac{21296}{81}\zeta_{2}-\frac{66110}{729}\biggr{)}l_{s}^{3}+\biggl{(}4\zeta_{2}-\frac{9493}{81}\biggr{)}l_{s}^{4}+\frac{986}{405}l_{s}^{5}\biggr{]}\frac{1}{\hat{s}^{2}}$
$\displaystyle+\biggl{[}320\zeta_{3}^{2}-\frac{1024}{3}l_{2}\zeta_{2}^{2}+\frac{83744}{35}\zeta_{2}^{3}-\frac{105920}{9}\zeta_{5}+\frac{108032}{81}l_{2}^{4}$
$\displaystyle-\frac{151808}{27}\zeta_{2}\zeta_{3}-\frac{216064}{27}l_{2}^{2}\zeta_{2}+\frac{591248}{243}\zeta_{3}+\frac{864256}{27}a_{4}$
$\displaystyle-\frac{5388448}{405}\zeta_{2}^{2}-\frac{36299776}{81}l_{2}\zeta_{2}-\frac{180817228}{6561}+\frac{320235272}{729}\zeta_{2}$
$\displaystyle+\biggl{(}-1280\zeta_{2}\zeta_{3}+4992\zeta_{5}+\frac{1024}{81}l_{2}^{4}-\frac{2048}{27}l_{2}^{2}\zeta_{2}+\frac{2336}{5}\zeta_{2}^{2}$
$\displaystyle+\frac{8192}{27}a_{4}-\frac{54272}{9}l_{2}\zeta_{2}-\frac{3323392}{81}\zeta_{3}+\frac{5432992}{243}\zeta_{2}+\frac{7229564}{81}\biggr{)}l_{s}$
$\displaystyle+\biggl{(}-\frac{416}{3}\zeta_{2}-\frac{2352}{5}\zeta_{2}^{2}-\frac{53728}{27}\zeta_{3}+\frac{4294144}{243}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}\frac{126080}{81}\zeta_{2}+\frac{64}{3}\zeta_{3}-\frac{296848}{243}\biggr{)}l_{s}^{3}+\biggl{(}24\zeta_{2}-\frac{141088}{243}\biggr{)}l_{s}^{4}$
$\displaystyle+\frac{5324}{405}l_{s}^{5}+\frac{34}{45}l_{s}^{6}\biggr{]}\frac{1}{\hat{s}^{3}}+O\left(\frac{1}{\hat{s}^{4}}\right),$
$\displaystyle F^{n_{h}}_{A,2}(\hat{s})$ $\displaystyle=$
$\displaystyle\biggl{[}384\zeta_{2}\zeta_{3}-320\zeta_{5}-\frac{256}{81}l_{2}^{4}+\frac{512}{27}l_{2}^{2}\zeta_{2}-\frac{2048}{27}a_{4}-\frac{4256}{27}\zeta_{3}$
(B.8)
$\displaystyle-\frac{67936}{135}\zeta_{2}^{2}+\frac{111296}{243}+\frac{327680}{27}l_{2}\zeta_{2}-\frac{329024}{27}\zeta_{2}$
$\displaystyle+\biggl{(}\frac{512}{3}\zeta_{3}-\frac{13312}{27}\zeta_{2}-\frac{637504}{243}\biggr{)}l_{s}$
$\displaystyle+\biggl{(}\frac{256}{3}\zeta_{2}-\frac{49168}{81}\biggr{)}l_{s}^{2}+\frac{832}{81}l_{s}^{3}+\frac{320}{27}l_{s}^{4}\biggr{]}\frac{1}{\hat{s}}$
$\displaystyle+\biggl{[}\frac{5504}{27}\zeta_{2}\zeta_{3}-\frac{9280}{3}-\frac{15872}{243}l_{2}^{4}+\frac{31744}{81}l_{2}^{2}\zeta_{2}-\frac{46720}{27}\zeta_{5}$
$\displaystyle-\frac{119584}{405}\zeta_{2}^{2}-\frac{126976}{81}a_{4}+\frac{506560}{81}\zeta_{3}+\frac{2017280}{27}l_{2}\zeta_{2}$
$\displaystyle-\frac{17172416}{243}\zeta_{2}+\biggl{(}\frac{306176}{81}\zeta_{3}-\frac{70592}{135}\zeta_{2}^{2}-\frac{110272}{81}\zeta_{2}$
$\displaystyle-\frac{2675584}{243}\biggr{)}l_{s}+\biggl{(}-\frac{7904}{81}\zeta_{2}-\frac{11648}{27}\zeta_{3}-\frac{576256}{243}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}-\frac{3520}{27}\zeta_{2}+\frac{6848}{27}\biggr{)}l_{s}^{3}+\frac{15904}{243}l_{s}^{4}+\frac{16}{81}l_{s}^{5}\biggr{]}\frac{1}{\hat{s}^{2}}$
$\displaystyle+\biggl{[}1408\zeta_{3}^{2}+\frac{1024}{3}l_{2}\zeta_{2}^{2}+\frac{9152}{105}\zeta_{2}^{3}+\frac{93952}{27}\zeta_{2}\zeta_{3}-\frac{126976}{243}l_{2}^{4}$
$\displaystyle+\frac{212224}{27}\zeta_{5}+\frac{251008}{135}\zeta_{2}^{2}+\frac{253952}{81}l_{2}^{2}\zeta_{2}+\frac{614656}{81}\zeta_{3}$
$\displaystyle-\frac{1015808}{81}a_{4}+\frac{8947712}{27}l_{2}\zeta_{2}+\frac{17539406}{2187}-\frac{79538816}{243}\zeta_{2}$
$\displaystyle+\biggl{(}2048\zeta_{2}\zeta_{3}-512\zeta_{5}-\frac{1024}{81}l_{2}^{4}+\frac{2048}{27}l_{2}^{2}\zeta_{2}+\frac{7616}{135}\zeta_{2}^{2}$
$\displaystyle-\frac{8192}{27}a_{4}+\frac{623488}{27}\zeta_{3}-\frac{3769600}{243}\zeta_{2}-\frac{48018980}{729}\biggr{)}l_{s}$
$\displaystyle+\biggl{(}\frac{3872}{5}\zeta_{2}^{2}+\frac{13952}{27}\zeta_{3}-\frac{174656}{81}\zeta_{2}-\frac{2940964}{243}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}-128\zeta_{3}-\frac{99008}{81}\zeta_{2}+\frac{324088}{243}\biggr{)}l_{s}^{3}-\biggl{(}16\zeta_{2}-\frac{115612}{243}\biggr{)}l_{s}^{4}$
$\displaystyle+\frac{824}{405}l_{s}^{5}+\frac{68}{45}l_{s}^{6}\biggr{]}\frac{1}{\hat{s}^{3}}+O\left(\frac{1}{\hat{s}^{4}}\right),$
$\displaystyle F^{n_{h}}_{S}(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{64}{3}\zeta_{2}^{2}l_{2}-\frac{107008}{81}\zeta_{2}l_{2}+\frac{7616}{81}\zeta_{2}l_{2}^{2}-\frac{3808}{243}l_{2}^{4}+\frac{29116}{135}\zeta_{2}^{2}-\frac{464}{3}\zeta_{2}\zeta_{3}$
(B.9)
$\displaystyle+\frac{177760}{243}\zeta_{2}-\frac{37876}{81}\zeta_{3}-\frac{4856}{9}\zeta_{5}+\frac{200698}{243}-\frac{30464}{81}a_{4}$
$\displaystyle+\biggl{(}-\frac{512}{27}a_{4}+\frac{128}{27}\zeta_{2}l_{2}^{2}-\frac{64}{81}l_{2}^{4}+\frac{368}{27}\zeta_{2}^{2}+\frac{20288}{243}\zeta_{2}-\frac{1420}{9}\zeta_{3}$
$\displaystyle+\frac{19496}{243}\biggr{)}l_{s}+\biggl{(}\frac{1472}{81}\zeta_{2}-\frac{160}{9}\zeta_{3}+\frac{16256}{243}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}-\frac{416}{81}\zeta_{2}+\frac{5624}{243}\biggr{)}l_{s}^{3}+\frac{104}{81}l_{s}^{4}-\frac{32}{81}l_{s}^{5}+\biggl{[}\frac{256}{9}\zeta_{2}l_{2}^{2}+\frac{78008}{81}\zeta_{2}^{2}$
$\displaystyle-\frac{128}{27}l_{2}^{4}-\frac{152320}{9}\zeta_{2}l_{2}-\frac{10528}{27}\zeta_{2}\zeta_{3}+\frac{131744}{9}\zeta_{2}-\frac{5080}{81}\zeta_{3}$
$\displaystyle-\frac{20192}{27}\zeta_{5}-\frac{4432}{3}-\frac{1024}{9}a_{4}+\biggl{(}\frac{108416}{81}\zeta_{2}-\frac{2512}{27}\zeta_{2}^{2}-\frac{59552}{81}\zeta_{3}$
$\displaystyle+\frac{203600}{81}\biggr{)}l_{s}+\biggl{(}-\frac{2600}{81}\zeta_{2}-\frac{2576}{27}\zeta_{3}+\frac{46400}{81}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}-\frac{32}{27}\zeta_{2}+\frac{8464}{81}\biggr{)}l_{s}^{3}-\frac{1280}{81}l_{s}^{4}\biggr{]}\frac{1}{\hat{s}}+\biggl{[}-\frac{22226279}{6561}+\frac{128}{3}\zeta_{2}^{2}l_{2}$
$\displaystyle-\frac{6406144}{81}\zeta_{2}l_{2}-\frac{44672}{81}\zeta_{2}l_{2}^{2}+\frac{22336}{243}l_{2}^{4}-\frac{2144}{35}\zeta_{2}^{3}+\frac{1023424}{405}\zeta_{2}^{2}$
$\displaystyle-\frac{43408}{27}\zeta_{2}\zeta_{3}+\frac{17358044}{243}\zeta_{2}+\frac{203516}{81}\zeta_{3}-\frac{105608}{27}\zeta_{5}+\frac{178688}{81}a_{4}$
$\displaystyle-272\zeta_{3}^{2}+\biggl{(}\frac{256}{27}\zeta_{2}l_{2}^{2}-\frac{1024}{27}a_{4}-\frac{19456}{27}\zeta_{2}l_{2}-\frac{128}{81}l_{2}^{4}-\frac{61216}{135}\zeta_{2}^{2}$
$\displaystyle+\frac{475480}{81}\zeta_{2}-\frac{328664}{81}\zeta_{3}-288\zeta_{2}\zeta_{3}-624\zeta_{5}+\frac{25638926}{2187}\biggr{)}l_{s}$
$\displaystyle+\biggl{(}-\frac{432}{5}\zeta_{2}^{2}+\frac{11944}{81}\zeta_{2}-\frac{3616}{9}\zeta_{3}+\frac{2984282}{729}\biggr{)}l_{s}^{2}+\biggl{(}-\frac{32}{3}\zeta_{3}$
$\displaystyle+\frac{1664}{27}\zeta_{2}+\frac{236572}{729}\biggr{)}l_{s}^{3}+\biggl{(}8\zeta_{2}-\frac{1424}{27}\biggr{)}l_{s}^{4}-\frac{832}{405}l_{s}^{5}-\frac{17}{45}l_{s}^{6}\biggr{]}\frac{1}{\hat{s}^{2}}$
$\displaystyle+\biggl{[}-\frac{95621945}{6561}+\frac{512}{3}\zeta_{2}^{2}l_{2}-\frac{28773376}{81}\zeta_{2}l_{2}-\frac{324608}{81}\zeta_{2}l_{2}^{2}$
$\displaystyle+\frac{162304}{243}l_{2}^{4}-\frac{12864}{35}\zeta_{2}^{3}+\frac{565208}{81}\zeta_{2}^{2}+\frac{241177964}{729}\zeta_{2}$
$\displaystyle-\frac{179552}{27}\zeta_{2}\zeta_{3}+\frac{4438880}{243}\zeta_{3}-\frac{59792}{3}\zeta_{5}-1632\zeta_{3}^{2}+\frac{1298432}{81}a_{4}$
$\displaystyle+\biggl{(}-\frac{4096}{27}a_{4}-\frac{152576}{27}\zeta_{2}l_{2}+\frac{1024}{27}\zeta_{2}l_{2}^{2}-\frac{512}{81}l_{2}^{4}-\frac{120976}{45}\zeta_{2}^{2}$
$\displaystyle+\frac{7481048}{243}\zeta_{2}-\frac{1568096}{81}\zeta_{3}-1728\zeta_{2}\zeta_{3}-3744\zeta_{5}+\frac{138016885}{2187}\biggr{)}l_{s}$
$\displaystyle+\biggl{(}-\frac{2592}{5}\zeta_{2}^{2}+\frac{68104}{81}\zeta_{2}-\frac{15776}{9}\zeta_{3}+\frac{14330044}{729}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}\frac{43600}{81}\zeta_{2}-64\zeta_{3}+\frac{870614}{729}\biggr{)}l_{s}^{3}+\biggl{(}48\zeta_{2}-\frac{83306}{243}\biggr{)}l_{s}^{4}$
$\displaystyle-\frac{2342}{135}l_{s}^{5}-\frac{34}{15}l_{s}^{6}\biggr{]}\frac{1}{\hat{s}^{3}}+O\left(\frac{1}{\hat{s}^{4}}\right),$
$\displaystyle F^{n_{h}}_{P}(\hat{s})$ $\displaystyle=$
$\displaystyle\frac{64}{3}\zeta_{2}^{2}l_{2}-\frac{107008}{81}\zeta_{2}l_{2}+\frac{7616}{81}\zeta_{2}l_{2}^{2}-\frac{3808}{243}l_{2}^{4}+\frac{29116}{135}\zeta_{2}^{2}-\frac{464}{3}\zeta_{2}\zeta_{3}$
(B.10)
$\displaystyle+\frac{177760}{243}\zeta_{2}-\frac{37876}{81}\zeta_{3}-\frac{4856}{9}\zeta_{5}+\frac{200698}{243}-\frac{30464}{81}a_{4}$
$\displaystyle+\biggl{(}-\frac{512}{27}a_{4}+\frac{128}{27}\zeta_{2}l_{2}^{2}-\frac{64}{81}l_{2}^{4}+\frac{368}{27}\zeta_{2}^{2}+\frac{20288}{243}\zeta_{2}-\frac{1420}{9}\zeta_{3}$
$\displaystyle+\frac{19496}{243}\biggr{)}l_{s}+\biggl{(}\frac{1472}{81}\zeta_{2}-\frac{160}{9}\zeta_{3}+\frac{16256}{243}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}-\frac{416}{81}\zeta_{2}+\frac{5624}{243}\biggr{)}l_{s}^{3}+\frac{104}{81}l_{s}^{4}-\frac{32}{81}l_{s}^{5}+\biggl{[}\frac{256}{27}\zeta_{2}l_{2}^{2}-\frac{121088}{27}\zeta_{2}l_{2}$
$\displaystyle-\frac{405008}{243}-\frac{128}{81}l_{2}^{4}+\frac{41224}{405}\zeta_{2}^{2}-\frac{5344}{27}\zeta_{2}\zeta_{3}+\frac{984992}{243}\zeta_{2}+\frac{49064}{81}\zeta_{3}$
$\displaystyle-\frac{13280}{27}\zeta_{5}-\frac{1024}{27}a_{4}+\biggl{(}-\frac{2512}{27}\zeta_{2}^{2}+\frac{5504}{9}\zeta_{2}-\frac{41504}{81}\zeta_{3}$
$\displaystyle+\frac{261968}{243}\biggr{)}l_{s}+\biggl{(}-\frac{1688}{81}\zeta_{2}-\frac{2576}{27}\zeta_{3}+\frac{25328}{243}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}-\frac{32}{27}\zeta_{2}+\frac{6224}{81}\biggr{)}l_{s}^{3}-\frac{640}{81}l_{s}^{4}\biggr{]}\frac{1}{\hat{s}}+\biggl{[}-\frac{25532903}{6561}+\frac{128}{3}\zeta_{2}^{2}l_{2}$
$\displaystyle-\frac{1576960}{81}\zeta_{2}l_{2}-\frac{37504}{81}\zeta_{2}l_{2}^{2}+\frac{18752}{243}l_{2}^{4}+\frac{2208}{7}\zeta_{2}^{3}-\frac{287296}{405}\zeta_{2}^{2}$
$\displaystyle-\frac{13072}{27}\zeta_{2}\zeta_{3}+\frac{1327348}{81}\zeta_{2}+\frac{27796}{27}\zeta_{3}-\frac{5896}{27}\zeta_{5}+\frac{150016}{81}a_{4}$
$\displaystyle+304\zeta_{3}^{2}+\biggl{(}-\frac{1024}{27}a_{4}-\frac{19456}{27}\zeta_{2}l_{2}+\frac{256}{27}\zeta_{2}l_{2}^{2}-\frac{128}{81}l_{2}^{4}+\frac{22976}{135}\zeta_{2}^{2}$
$\displaystyle+\frac{62744}{81}\zeta_{2}-\frac{190744}{81}\zeta_{3}+224\zeta_{2}\zeta_{3}+528\zeta_{5}+\frac{5114702}{2187}\biggr{)}l_{s}$
$\displaystyle+\biggl{(}\frac{432}{5}\zeta_{2}^{2}-\frac{36152}{81}\zeta_{2}-\frac{2048}{9}\zeta_{3}+\frac{701306}{729}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}-\frac{128}{3}\zeta_{2}-\frac{64}{3}\zeta_{3}+\frac{179260}{729}\biggr{)}l_{s}^{3}+\frac{424}{243}l_{s}^{4}+\frac{1192}{405}l_{s}^{5}+\frac{17}{45}l_{s}^{6}\biggr{]}\frac{1}{\hat{s}^{2}}$
$\displaystyle+\biggl{[}-\frac{85198093}{6561}+\frac{512}{3}\zeta_{2}^{2}l_{2}-\frac{5825536}{81}\zeta_{2}l_{2}-\frac{26624}{27}\zeta_{2}l_{2}^{2}+\frac{13312}{81}l_{2}^{4}$
$\displaystyle+\frac{4416}{7}\zeta_{2}^{3}+\frac{101672}{405}\zeta_{2}^{2}-\frac{44192}{27}\zeta_{2}\zeta_{3}+\frac{38327036}{729}\zeta_{2}+\frac{364016}{243}\zeta_{3}$
$\displaystyle+\frac{4400}{9}\zeta_{5}+608\zeta_{3}^{2}+\frac{106496}{27}a_{4}+\biggl{(}1056\zeta_{5}-\frac{4096}{27}a_{4}-\frac{5120}{3}\zeta_{2}l_{2}$
$\displaystyle+\frac{1024}{27}\zeta_{2}l_{2}^{2}-\frac{512}{81}l_{2}^{4}+\frac{9232}{9}\zeta_{2}^{2}+\frac{2008}{243}\zeta_{2}-\frac{532288}{81}\zeta_{3}+448\zeta_{2}\zeta_{3}$
$\displaystyle+\frac{5075447}{729}\biggr{)}l_{s}+\biggl{(}\frac{864}{5}\zeta_{2}^{2}-\frac{36776}{27}\zeta_{2}-\frac{25792}{27}\zeta_{3}+\frac{965428}{243}\biggr{)}l_{s}^{2}$
$\displaystyle+\biggl{(}-\frac{17584}{81}\zeta_{2}-\frac{128}{3}\zeta_{3}+\frac{235522}{243}\biggr{)}l_{s}^{3}+\frac{10862}{243}l_{s}^{4}+\frac{6254}{405}l_{s}^{5}$
$\displaystyle+\frac{34}{45}l_{s}^{6}\biggr{]}\frac{1}{\hat{s}^{3}}+O\left(\frac{1}{\hat{s}^{4}}\right).$

##  Appendix C Partially resummed expansions at $x=0$

In Eq. (3.69) we showed the result for the partially resummed expansion of
$F^{(0)}_{V,1,1}(x)+\zeta_{2}F^{(0)}_{V,1,2}(x)+\zeta_{3}F^{(0)}_{V,1,3}(x)$.
In this appendix, we show the corresponding results for the other form
factors,

$\displaystyle
F^{(0)}_{V,2,1}(x)+\zeta_{2}F^{(0)}_{V,2,2}(x)+\zeta_{3}F^{(0)}_{V,2,3}(x)=$
(C.1)
$\displaystyle\ln(2)\Biggl{\\{}\frac{1024x\zeta_{2}^{2}P_{48}}{9(x-1)^{3}(1+x)^{5}}+\Biggl{[}\frac{256x\ln(1+x)P_{18}}{9(x-1)^{2}(1+x)^{4}}+\frac{512x{\rm
Li}_{2}(-x)P_{46}}{9(x-1)^{3}(1+x)^{5}}$
$\displaystyle-\frac{256x^{2}\ln^{2}(x)P_{49}}{9(x-1)^{3}(1+x)^{6}}-\frac{64P_{66}}{81(x-1)^{2}(1+x)^{6}}$
$\displaystyle-\ln(1-x)\Biggl{(}\frac{1024x^{2}\ln(x)P_{1}}{9(x-1)^{3}(1+x)^{5}}-\frac{1024xP_{4}}{9(x-1)^{2}(1+x)^{4}}\Biggr{)}$
$\displaystyle+\ln(x)\Biggl{(}\frac{512x\ln(1+x)P_{47}}{9(x-1)^{3}(1+x)^{5}}-\frac{256xP_{55}}{27(x-1)^{3}(1+x)^{5}}\Biggr{)}+\frac{256x^{2}\big{(}1-7x+x^{2}\big{)}\ln^{3}(x)}{(x-1)(1+x)^{5}}$
$\displaystyle-\frac{1024x^{2}\big{(}1-12x+x^{2}\big{)}\big{(}1-x+x^{2}\big{)}{\rm
Li}_{2}(x)}{9(x-1)^{3}(1+x)^{5}}\Biggr{]}\zeta_{2}\Biggr{\\}}+\ln^{2}(2)\Biggl{\\{}\Biggr{[}\frac{256x^{2}\ln(x)P_{5}}{9(x-1)^{3}(1+x)^{5}}$
$\displaystyle-\frac{128xP_{60}}{81(x-1)^{3}(1+x)^{5}}-\ln(1-x)\Biggl{(}-\frac{1024x\big{(}1-x+x^{2}\big{)}}{(x-1)(1+x)^{3}}-\frac{1024x^{2}\big{(}1-x+x^{2}\big{)}\ln(x)}{(x-1)(1+x)^{5}}\Biggr{)}$
$\displaystyle-\frac{256x^{2}\big{(}1-x+x^{2}\big{)}\ln^{2}(x)}{(x-1)(1+x)^{5}}-\frac{1024x\big{(}1-x+x^{2}\big{)}\ln(1+x)}{(x-1)(1+x)^{3}}+\frac{1024x^{2}\big{(}1-x+x^{2}\big{)}{\rm
Li}_{2}(x)}{(x-1)(1+x)^{5}}\Biggr{]}\zeta_{2}$
$\displaystyle-\frac{1024x^{2}\big{(}1-x+x^{2}\big{)}\zeta_{2}^{2}}{(x-1)(1+x)^{5}}\Biggr{\\}}+{\rm
Li}_{4}\left(\frac{1}{2}\right)\Biggl{\\{}-\frac{1024x^{2}\ln(x)P_{11}}{9(x-1)^{3}(1+x)^{5}}+\frac{1024xP_{56}}{81(x-1)^{3}(1+x)^{5}}$
$\displaystyle-\ln(1-x)\Biggl{[}\frac{4096x\big{(}1-x+x^{2}\big{)}}{(x-1)(1+x)^{3}}+\frac{4096x^{2}\big{(}1-x+x^{2}\big{)}\ln(x)}{(x-1)(1+x)^{5}}\Biggr{]}$
$\displaystyle+\frac{1024x^{2}\big{(}1-x+x^{2}\big{)}\ln^{2}(x)}{(x-1)(1+x)^{5}}+\frac{4096x\big{(}1-x+x^{2}\big{)}\ln(1+x)}{(x-1)(1+x)^{3}}-\frac{4096x^{2}\big{(}1-x+x^{2}\big{)}{\rm
Li}_{2}(x)}{(x-1)(1+x)^{5}}\Biggr{\\}}$
$\displaystyle+\ln^{4}(2)\Biggl{\\{}-\frac{128x^{2}\ln(x)P_{11}}{27(x-1)^{3}(1+x)^{5}}+\frac{128xP_{56}}{243(x-1)^{3}(1+x)^{5}}-\ln(1-x)\Biggl{[}\frac{512x\big{(}1-x+x^{2}\big{)}}{3(x-1)(1+x)^{3}}$
$\displaystyle+\frac{512x^{2}\big{(}1-x+x^{2}\big{)}\ln(x)}{3(x-1)(1+x)^{5}}\Biggr{]}+\frac{512x^{2}\big{(}1-x+x^{2}\big{)}\zeta_{2}}{3(x-1)(1+x)^{5}}+\frac{128x^{2}\big{(}1-x+x^{2}\big{)}\ln^{2}(x)}{3(x-1)(1+x)^{5}}$
$\displaystyle+\frac{512x\big{(}1-x+x^{2}\big{)}\ln(1+x)}{3(x-1)(1+x)^{3}}-\frac{512x^{2}\big{(}1-x+x^{2}\big{)}{\rm
Li}_{2}(x)}{3(x-1)(1+x)^{5}}\Biggr{\\}}+\Biggl{\\{}\frac{4096{\rm
Li}_{4}\left(\frac{1}{2}\right)x^{2}\big{(}1-x+x^{2}\big{)}}{(x-1)(1+x)^{5}}$
$\displaystyle+\frac{768x^{3}\big{(}1+x^{2}\big{)}\zeta_{3}\ln(x)}{(x-1)^{3}(1+x)^{5}}-\frac{16x^{3}\ln^{4}(x)}{(x-1)^{3}(1+x)^{3}}\Biggr{\\}}\zeta_{2}+\frac{64x^{2}\big{(}1-4x+x^{2}\big{)}^{2}\zeta_{3}\ln^{3}(x)}{3(x-1)^{3}(1+x)^{5}}$
$\displaystyle-\frac{68x^{3}\big{(}1-4x+x^{2}\big{)}\ln^{6}(x)}{45(x-1)^{3}(1+x)^{5}}-\frac{128x^{2}\zeta_{3}^{2}P_{8}}{(x-1)^{3}(1+x)^{5}}+\frac{128x^{2}\zeta_{5}\ln(x)P_{10}}{(x-1)^{3}(1+x)^{5}}-\frac{32x^{2}\zeta_{2}^{2}\ln^{2}(x)P_{24}}{5(x-1)^{3}(1+x)^{5}}$
$\displaystyle-\frac{64x^{2}\zeta_{2}^{3}P_{38}}{105(x-1)^{3}(1+x)^{5}}+\tilde{F}^{(0),\rm
rest}_{V,2}(x),$ $\displaystyle
F^{(0)}_{A,1,1}(x)+\zeta_{2}F^{(0)}_{A,1,2}(x)+\zeta_{3}F^{(0)}_{A,1,3}(x)=$
(C.2)
$\displaystyle\ln(2)\Biggl{\\{}\frac{256\big{(}1+x^{2}\big{)}\zeta_{2}^{2}P_{27}}{9(x-1)^{3}(1+x)^{3}}+\Biggl{[}\frac{128\ln(1+x)P_{15}}{9(x-1)^{2}(1+x)^{2}}+\frac{64\big{(}1+x^{2}\big{)}{\rm
Li}_{2}(-x)P_{36}}{9(x-1)^{3}(1+x)^{3}}$ $\displaystyle+\frac{64{\rm
Li}_{2}(x)P_{51}}{9(x-1)^{3}(1+x)^{3}}-\frac{64\ln^{2}(x)P_{61}}{9(x-1)^{3}(1+x)^{4}}$
$\displaystyle-\frac{16P_{83}}{81(x-1)^{4}x(1+x)^{6}}-\ln(1-x)\Biggl{(}\frac{128\big{(}3-2x^{2}+3x^{4}\big{)}}{9(x-1)^{2}(1+x)^{2}}$
$\displaystyle-\frac{64\ln(x)P_{52}}{9(x-1)^{3}(1+x)^{3}}\Biggr{)}+\ln(x)\Biggl{(}\frac{64\ln(1+x)P_{57}}{9(x-1)^{3}(1+x)^{3}}-\frac{32P_{79}}{27(x-1)^{5}(1+x)^{5}}\Biggr{)}$
$\displaystyle-\frac{192x^{2}\ln^{3}(x)}{(x-1)(1+x)^{3}}\Biggr{]}\zeta_{2}\Biggr{\\}}+\ln^{2}(2)\Biggl{\\{}\Biggl{[}\frac{128\ln(x)P_{44}}{9(x-1)^{3}(1+x)^{3}}-\frac{32P_{74}}{81(x-1)^{5}(1+x)^{5}}$
$\displaystyle-\ln(1-x)\Biggl{(}-\frac{128\big{(}1+x^{2}\big{)}\big{(}11-2x+11x^{2}\big{)}}{3(x-1)(1+x)^{3}}+\frac{256x^{2}\ln(x)}{(x-1)(1+x)^{3}}\Biggr{)}+\frac{64x^{2}\ln^{2}(x)}{(x-1)(1+x)^{3}}$
$\displaystyle-\frac{128\big{(}1+x^{2}\big{)}\big{(}11-2x+11x^{2}\big{)}\ln(1+x)}{3(x-1)(1+x)^{3}}-\frac{256x^{2}{\rm
Li}_{2}(x)}{(x-1)(1+x)^{3}}\Biggr{]}\zeta_{2}+\frac{256x^{2}\zeta_{2}^{2}}{(x-1)(1+x)^{3}}\Biggr{\\}}$
$\displaystyle+\ln^{4}(2)\Biggl{\\{}-\frac{64\ln(x)P_{42}}{27(x-1)^{3}(1+x)^{3}}-\frac{16P_{67}}{243(x-1)^{5}(1+x)^{5}}-\ln(1-x)$
$\displaystyle\times\Biggl{[}\frac{64\big{(}1+x^{2}\big{)}\big{(}11-2x+11x^{2}\big{)}}{9(x-1)(1+x)^{3}}-\frac{128x^{2}\ln(x)}{3(x-1)(1+x)^{3}}\Biggr{]}-\frac{128x^{2}\zeta_{2}}{3(x-1)(1+x)^{3}}$
$\displaystyle-\frac{32x^{2}\ln^{2}(x)}{3(x-1)(1+x)^{3}}+\frac{64\big{(}1+x^{2}\big{)}\big{(}11-2x+11x^{2}\big{)}\ln(1+x)}{9(x-1)(1+x)^{3}}+\frac{128x^{2}{\rm
Li}_{2}(x)}{3(x-1)(1+x)^{3}}\Biggr{\\}}$ $\displaystyle+{\rm
Li}_{4}\left(\frac{1}{2}\right)\Biggl{\\{}-\frac{512\ln(x)P_{42}}{9(x-1)^{3}(1+x)^{3}}-\frac{128P_{67}}{81(x-1)^{5}(1+x)^{5}}-\ln(1-x)$
$\displaystyle\times\Biggl{[}\frac{512\big{(}1+x^{2}\big{)}\big{(}11-2x+11x^{2}\big{)}}{3(x-1)(1+x)^{3}}-\frac{1024x^{2}\ln(x)}{(x-1)(1+x)^{3}}\Biggr{]}-\frac{256x^{2}\ln^{2}(x)}{(x-1)(1+x)^{3}}$
$\displaystyle+\frac{512\big{(}1+x^{2}\big{)}\big{(}11-2x+11x^{2}\big{)}\ln(1+x)}{3(x-1)(1+x)^{3}}+\frac{1024x^{2}{\rm
Li}_{2}(x)}{(x-1)(1+x)^{3}}\Biggr{\\}}+\Biggl{\\{}-\frac{1024{\rm
Li}_{4}\left(\frac{1}{2}\right)x^{2}}{(x-1)(1+x)^{3}}$
$\displaystyle+\frac{128x^{2}\big{(}1+5x+x^{2}\big{)}\zeta_{3}\ln(x)}{(x-1)^{3}(1+x)^{3}}-\frac{4x^{2}\ln^{4}(x)}{(x-1)(1+x)^{3}}\Biggr{\\}}\zeta_{2}+\frac{16x^{2}\big{(}3+32x+3x^{2}\big{)}\zeta_{3}^{2}}{(x-1)^{3}(1+x)^{3}}$
$\displaystyle-\frac{16x^{2}\big{(}2011-8162x+2011x^{2}\big{)}\zeta_{2}^{3}}{105(x-1)^{3}(1+x)^{3}}-\frac{576x^{2}\big{(}1-3x+x^{2}\big{)}\zeta_{5}\ln(x)}{(x-1)^{3}(1+x)^{3}}$
$\displaystyle+\frac{8x^{2}\big{(}65-22x+65x^{2}\big{)}\zeta_{2}^{2}\ln^{2}(x)}{5(x-1)^{3}(1+x)^{3}}-\frac{32x^{2}\zeta_{3}\ln^{3}(x)}{3(x-1)^{3}(1+x)}+\frac{34x^{3}\ln^{6}(x)}{45(x-1)^{3}(1+x)^{3}}$
$\displaystyle+\tilde{F}^{(0),\rm rest}_{A,1}(x),$ $\displaystyle
F^{(0)}_{A,2,1}(x)+\zeta_{2}F^{(0)}_{A,2,2}(x)+\zeta_{3}F^{(0)}_{A,2,3}(x)=$
(C.3)
$\displaystyle\ln(2)\Biggl{\\{}-\frac{2048x^{2}\zeta_{2}^{2}P_{29}}{9(x-1)^{5}(1+x)^{3}}+\Biggl{[}\frac{1024x^{2}{\rm
Li}_{2}(x)P_{2}}{9(x-1)^{5}(1+x)^{3}}+\frac{256x\ln(1+x)P_{14}}{9(x-1)^{4}(1+x)^{2}}$
$\displaystyle-\frac{512x^{2}{\rm
Li}_{2}(-x)P_{32}}{9(x-1)^{5}(1+x)^{3}}+\frac{256x^{2}\ln^{2}(x)P_{39}}{9(x-1)^{5}(1+x)^{4}}+\frac{64P_{75}}{27(x-1)^{4}(1+x)^{6}}$
$\displaystyle-\ln(1-x)\Biggl{(}\frac{1024x^{2}\big{(}1+4x+x^{2}\big{)}}{9(x-1)^{4}(1+x)^{2}}-\frac{1024x^{2}\ln(x)P_{3}}{9(x-1)^{5}(1+x)^{3}}\Biggr{)}+\ln(x)$
$\displaystyle\times\Biggl{(}-\frac{512x^{2}\ln(1+x)P_{33}}{9(x-1)^{5}(1+x)^{3}}-\frac{256xP_{64}}{27(x-1)^{5}(1+x)^{5}}\Biggr{)}-\frac{768x^{3}\ln^{3}(x)}{(x-1)^{3}(1+x)^{3}}\Biggr{]}\zeta_{2}\Biggr{\\}}$
$\displaystyle+\ln^{2}(2)\Biggl{\\{}\frac{256x^{2}\ln(x)P_{26}}{9(x-1)^{5}(1+x)^{3}}-\frac{128xP_{63}}{81(x-1)^{5}(1+x)^{5}}-\ln(1-x)$
$\displaystyle\times\Biggl{[}\frac{1024x^{2}\big{(}3-2x+3x^{2}\big{)}}{(x-1)^{3}(1+x)^{3}}+\frac{1024x^{3}\ln(x)}{(x-1)^{3}(1+x)^{3}}\Biggr{]}+\frac{256x^{3}\ln^{2}(x)}{(x-1)^{3}(1+x)^{3}}$
$\displaystyle+\frac{1024x^{2}\big{(}3-2x+3x^{2}\big{)}\ln(1+x)}{(x-1)^{3}(1+x)^{3}}-\frac{1024x^{3}{\rm
Li}_{2}(x)}{(x-1)^{3}(1+x)^{3}}\Biggr{]}\zeta_{2}+\frac{1024x^{3}\zeta_{2}^{2}}{(x-1)^{3}(1+x)^{3}}\Biggr{\\}}$
$\displaystyle+\ln^{4}(2)\Biggl{\\{}-\frac{128x^{2}\ln(x)P_{23}}{27(x-1)^{5}(1+x)^{3}}-\frac{128xP_{62}}{243(x-1)^{5}(1+x)^{5}}-\ln(1-x)$
$\displaystyle\times\Biggl{[}-\frac{512x^{2}\big{(}3-2x+3x^{2}\big{)}}{3(x-1)^{3}(1+x)^{3}}-\frac{512x^{3}\ln(x)}{3(x-1)^{3}(1+x)^{3}}\Biggr{]}$
$\displaystyle-\frac{512x^{3}\zeta_{2}}{3(x-1)^{3}(1+x)^{3}}-\frac{128x^{3}\ln^{2}(x)}{3(x-1)^{3}(1+x)^{3}}-\frac{512x^{2}\big{(}3-2x+3x^{2}\big{)}\ln(1+x)}{3(x-1)^{3}(1+x)^{3}}$
$\displaystyle+\frac{512x^{3}{\rm
Li}_{2}(x)}{3(x-1)^{3}(1+x)^{3}}\Biggr{\\}}+{\rm
Li}_{4}\left(\frac{1}{2}\right)\Biggl{\\{}-\frac{1024x^{2}\ln(x)P_{23}}{9(x-1)^{5}(1+x)^{3}}-\frac{1024xP_{62}}{81(x-1)^{5}(1+x)^{5}}$
$\displaystyle-\ln(1-x)\Biggl{[}-\frac{4096x^{2}\big{(}3-2x+3x^{2}\big{)}}{(x-1)^{3}(1+x)^{3}}-\frac{4096x^{3}\ln(x)}{(x-1)^{3}(1+x)^{3}}\Biggr{]}-\frac{1024x^{3}\ln^{2}(x)}{(x-1)^{3}(1+x)^{3}}$
$\displaystyle-\frac{4096x^{2}\big{(}3-2x+3x^{2}\big{)}\ln(1+x)}{(x-1)^{3}(1+x)^{3}}+\frac{4096x^{3}{\rm
Li}_{2}(x)}{(x-1)^{3}(1+x)^{3}}\Biggr{\\}}$
$\displaystyle+\Biggl{\\{}-\frac{4096{\rm
Li}_{4}\left(\frac{1}{2}\right)x^{3}}{(x-1)^{3}(1+x)^{3}}+\frac{768x^{3}\big{(}3+8x+3x^{2}\big{)}\zeta_{3}\ln(x)}{(x-1)^{5}(1+x)^{3}}-\frac{16x^{3}\ln^{4}(x)}{(x-1)^{3}(1+x)^{3}}\Biggr{\\}}\zeta_{2}$
$\displaystyle+\frac{64x^{3}\big{(}59+12302x+59x^{2}\big{)}\zeta_{2}^{3}}{105(x-1)^{5}(1+x)^{3}}+\frac{128x^{3}\big{(}11+35x+11x^{2}\big{)}\zeta_{3}^{2}}{(x-1)^{5}(1+x)^{3}}$
$\displaystyle-\frac{1152x^{3}\big{(}1-8x+x^{2}\big{)}\zeta_{5}\ln(x)}{(x-1)^{5}(1+x)^{3}}+\frac{32x^{3}\big{(}119+86x+119x^{2}\big{)}\zeta_{2}^{2}\ln^{2}(x)}{5(x-1)^{5}(1+x)^{3}}$
$\displaystyle-\frac{128x^{3}\zeta_{3}\ln^{3}(x)}{(x-1)^{5}(1+x)}+\frac{68x^{3}\big{(}1+4x+x^{2}\big{)}\ln^{6}(x)}{45(x-1)^{5}(1+x)^{3}}+\tilde{F}^{(0),\rm
rest}_{A,2}(x),$ $\displaystyle
F^{(0)}_{S,1}(x)+\zeta_{2}F^{(0)}_{S,2}(x)+\zeta_{3}F^{(0)}_{S,3}(x)=$ (C.4)
$\displaystyle\ln^{4}(2)\Biggl{\\{}-\frac{32\ln(x)P_{7}}{27(x-1)(1+x)^{3}}-\frac{64\ln(1-x)P_{19}}{9(x-1)(1+x)^{3}}+\frac{64\ln(1+x)P_{19}}{9(x-1)(1+x)^{3}}$
$\displaystyle+\frac{16P_{71}}{243(x-1)^{5}(1+x)^{5}}\Biggr{\\}}+{\rm
Li}_{4}\left(\frac{1}{2}\right)\Biggl{\\{}-\frac{256\ln(x)P_{7}}{9(x-1)(1+x)^{3}}-\frac{512\ln(1-x)P_{19}}{3(x-1)(1+x)^{3}}$
$\displaystyle+\frac{512\ln(1+x)P_{19}}{3(x-1)(1+x)^{3}}+\frac{128P_{71}}{81(x-1)^{5}(1+x)^{5}}\Biggr{\\}}+\ln(2)\Biggl{\\{}\frac{256\zeta_{2}^{2}P_{20}}{3(x-1)(1+x)^{3}}$
$\displaystyle+\Biggl{[}\frac{64{\rm
Li}_{2}(x)P_{31}}{9(x-1)(1+x)^{3}}+\frac{64{\rm
Li}_{2}(-x)P_{37}}{9(x-1)(1+x)^{3}}-\frac{64\ln^{2}(x)P_{41}}{9(x-1)^{2}(1+x)^{3}}$
$\displaystyle-\frac{64P_{80}}{81(x-1)^{4}x(1+x)^{6}}-\ln(1-x)\Biggl{(}\frac{128\big{(}3+8x+3x^{2}\big{)}}{9(1+x)^{2}}-\frac{64\ln(x)P_{22}}{3(x-1)(1+x)^{3}}\Biggr{)}$
$\displaystyle+\ln(x)\Biggl{(}\frac{64\ln(1+x)P_{25}}{3(x-1)(1+x)^{3}}-\frac{32P_{76}}{27(x-1)^{5}(1+x)^{5}}\Biggr{)}+\frac{64\big{(}9-32x+9x^{2}\big{)}\ln(1+x)}{9(1+x)^{2}}\Biggr{]}\zeta_{2}\Biggr{\\}}$
$\displaystyle+\ln(2)^{2}\Biggl{\\{}\frac{64\ln(x)P_{13}}{9(x-1)(1+x)^{3}}+\frac{128\ln(1-x)P_{19}}{3(x-1)(1+x)^{3}}-\frac{128\ln(1+x)P_{19}}{3(x-1)(1+x)^{3}}$
$\displaystyle-\frac{64P_{69}}{81(x-1)^{5}(1+x)^{5}}\Biggr{\\}}\zeta_{2}+\Biggl{\\{}\frac{192x^{2}\zeta_{3}\ln(x)}{(x-1)(1+x)^{3}}-\frac{8x^{2}\ln^{4}(x)}{(x-1)(1+x)^{3}}\Biggr{\\}}\zeta_{2}$
$\displaystyle+\frac{2144x^{2}\zeta_{2}^{3}}{35(x-1)(1+x)^{3}}+\frac{272x^{2}\zeta_{3}^{2}}{(x-1)(1+x)^{3}}+\frac{544x^{2}\zeta_{5}\ln(x)}{(x-1)(1+x)^{3}}+\frac{432x^{2}\zeta_{2}^{2}\ln^{2}(x)}{5(x-1)(1+x)^{3}}$
$\displaystyle+\frac{32x^{2}\zeta_{3}\ln^{3}(x)}{3(x-1)(1+x)^{3}}+\frac{17x^{2}\ln^{6}(x)}{45(x-1)(1+x)^{3}}+\tilde{F}^{(0),\rm
rest}_{S}(x),$ $\displaystyle
F^{(0)}_{P,1}(x)+\zeta_{2}F^{(0)}_{P,2}(x)+\zeta_{3}F^{(0)}_{P,3}(x)=$ (C.5)
$\displaystyle\ln(2)\Biggl{\\{}\frac{256\zeta_{2}^{2}P_{28}}{9(x-1)^{3}(1+x)}+\Biggl{[}\frac{64\ln(1+x)P_{16}}{9(x-1)^{2}(1+x)^{2}}+\frac{64{\rm
Li}_{2}(x)P_{21}}{3(x-1)^{3}(1+x)}+\frac{64{\rm
Li}_{2}(-x)P_{35}}{9(x-1)^{3}(1+x)}$
$\displaystyle-\frac{64\ln^{2}(x)P_{40}}{9(x-1)^{3}(1+x)^{2}}-\frac{64P_{81}}{81(x-1)^{4}x(1+x)^{6}}-\ln(1-x)\Biggl{(}\frac{128\big{(}3-8x+3x^{2}\big{)}}{9(x-1)^{2}}$
$\displaystyle-\frac{64\ln(x)P_{30}}{9(x-1)^{3}(1+x)}\Biggr{)}+\ln(x)\Biggl{(}\frac{64\ln(1+x)P_{34}}{9(x-1)^{3}(1+x)}-\frac{32P_{77}}{27(x-1)^{5}(1+x)^{5}}\Biggr{)}\Biggr{]}\zeta_{2}\Biggr{\\}}$
$\displaystyle+\ln^{4}(2)\Biggl{\\{}-\frac{32\ln(x)P_{6}}{27(x-1)^{3}(1+x)}+\frac{16P_{72}}{243(x-1)^{5}(1+x)^{5}}-\frac{64\big{(}11-6x+11x^{2}\big{)}\ln(1-x)}{9(x-1)(1+x)}$
$\displaystyle+\frac{64\big{(}11-6x+11x^{2}\big{)}\ln(1+x)}{9(x-1)(1+x)}\Biggr{\\}}+{\rm
Li}_{4}\left(\frac{1}{2}\right)\Biggl{\\{}-\frac{256\ln(x)P_{6}}{9(x-1)^{3}(1+x)}+\frac{128P_{72}}{81(x-1)^{5}(1+x)^{5}}$
$\displaystyle-\frac{512\big{(}11-6x+11x^{2}\big{)}\ln(1-x)}{3(x-1)(1+x)}+\frac{512\big{(}11-6x+11x^{2}\big{)}\ln(1+x)}{3(x-1)(1+x)}\Biggr{\\}}$
$\displaystyle+\ln^{2}(2)\Biggl{\\{}\frac{64\ln(x)P_{12}}{9(x-1)^{3}(1+x)}-\frac{64P_{70}}{81(x-1)^{5}(1+x)^{5}}+\frac{128\big{(}11-6x+11x^{2}\big{)}\ln(1-x)}{3(x-1)(1+x)}$
$\displaystyle-\frac{128\big{(}11-6x+11x^{2}\big{)}\ln(1+x)}{3(x-1)(1+x)}\Biggr{\\}}\zeta_{2}-\frac{2208x^{2}\zeta_{2}^{3}}{7(x-1)^{3}(1+x)}-\frac{304x^{2}\zeta_{3}^{2}}{(x-1)^{3}(1+x)}$
$\displaystyle-\frac{448x^{2}\zeta_{2}\zeta_{3}\ln(x)}{(x-1)^{3}(1+x)}-\frac{288x^{2}\zeta_{5}\ln(x)}{(x-1)^{3}(1+x)}-\frac{432x^{2}\zeta_{2}^{2}\ln^{2}(x)}{5(x-1)^{3}(1+x)}+\frac{64x^{2}\zeta_{3}\ln^{3}(x)}{3(x-1)^{3}(1+x)}$
$\displaystyle-\frac{17x^{2}\ln^{6}(x)}{45(x-1)^{3}(1+x)}+\tilde{F}^{(0),\rm
rest}_{P}(x).$

The polynomials are given by

$\displaystyle P_{1}$ $\displaystyle=$ $\displaystyle
x^{4}-21x^{3}+34x^{2}-21x+1,$ (C.6) $\displaystyle P_{2}$ $\displaystyle=$
$\displaystyle x^{4}-5x^{3}-10x^{2}-5x+1,$ (C.7) $\displaystyle P_{3}$
$\displaystyle=$ $\displaystyle x^{4}-x^{3}-6x^{2}-x+1,$ (C.8) $\displaystyle
P_{4}$ $\displaystyle=$ $\displaystyle x^{4}+5x^{3}-10x^{2}+5x+1,$ (C.9)
$\displaystyle P_{5}$ $\displaystyle=$ $\displaystyle
x^{4}+90x^{3}+46x^{2}+90x+1,$ (C.10) $\displaystyle P_{6}$ $\displaystyle=$
$\displaystyle 2x^{4}-13x^{3}+6x^{2}-13x+2,$ (C.11) $\displaystyle P_{7}$
$\displaystyle=$ $\displaystyle 2x^{4}-5x^{3}-2x^{2}-5x+2,$ (C.12)
$\displaystyle P_{8}$ $\displaystyle=$ $\displaystyle
2x^{4}+4x^{3}-29x^{2}+4x+2,$ (C.13) $\displaystyle P_{9}$ $\displaystyle=$
$\displaystyle 3x^{4}+16x^{3}+62x^{2}+16x+3,$ (C.14) $\displaystyle P_{10}$
$\displaystyle=$ $\displaystyle 4x^{4}-71x^{3}+168x^{2}-71x+4,$ (C.15)
$\displaystyle P_{11}$ $\displaystyle=$ $\displaystyle
7x^{4}+36x^{3}+106x^{2}+36x+7,$ (C.16) $\displaystyle P_{12}$ $\displaystyle=$
$\displaystyle 8x^{4}-25x^{3}+24x^{2}-25x+8,$ (C.17) $\displaystyle P_{13}$
$\displaystyle=$ $\displaystyle 8x^{4}+7x^{3}-8x^{2}+7x+8,$ (C.18)
$\displaystyle P_{14}$ $\displaystyle=$ $\displaystyle
9x^{4}-32x^{3}+34x^{2}-32x+9,$ (C.19) $\displaystyle P_{15}$ $\displaystyle=$
$\displaystyle 9x^{4}-27x^{3}+34x^{2}-27x+9,$ (C.20) $\displaystyle P_{16}$
$\displaystyle=$ $\displaystyle 9x^{4}-22x^{3}+34x^{2}-22x+9,$ (C.21)
$\displaystyle P_{17}$ $\displaystyle=$ $\displaystyle
9x^{4}-5x^{3}+98x^{2}-5x+9,$ (C.22) $\displaystyle P_{18}$ $\displaystyle=$
$\displaystyle 11x^{4}-56x^{3}+118x^{2}-56x+11,$ (C.23) $\displaystyle P_{19}$
$\displaystyle=$ $\displaystyle 11x^{4}+4x^{3}+34x^{2}+4x+11,$ (C.24)
$\displaystyle P_{20}$ $\displaystyle=$ $\displaystyle
12x^{4}-3x^{3}+41x^{2}-3x+12,$ (C.25) $\displaystyle P_{21}$ $\displaystyle=$
$\displaystyle 19x^{4}-38x^{3}+42x^{2}-38x+19,$ (C.26) $\displaystyle P_{22}$
$\displaystyle=$ $\displaystyle 19x^{4}+38x^{3}+34x^{2}+38x+19,$ (C.27)
$\displaystyle P_{23}$ $\displaystyle=$ $\displaystyle
29x^{4}-28x^{3}+94x^{2}-28x+29,$ (C.28) $\displaystyle P_{24}$
$\displaystyle=$ $\displaystyle 31x^{4}-141x^{3}+112x^{2}-141x+31,$ (C.29)
$\displaystyle P_{25}$ $\displaystyle=$ $\displaystyle
33x^{4}+12x^{3}+98x^{2}+12x+33,$ (C.30) $\displaystyle P_{26}$
$\displaystyle=$ $\displaystyle 35x^{4}-58x^{3}+106x^{2}-58x+35,$ (C.31)
$\displaystyle P_{27}$ $\displaystyle=$ $\displaystyle
36x^{4}-107x^{3}+141x^{2}-107x+36,$ (C.32) $\displaystyle P_{28}$
$\displaystyle=$ $\displaystyle 36x^{4}-99x^{3}+127x^{2}-99x+36,$ (C.33)
$\displaystyle P_{29}$ $\displaystyle=$ $\displaystyle
40x^{4}-106x^{3}+135x^{2}-106x+40,$ (C.34) $\displaystyle P_{30}$
$\displaystyle=$ $\displaystyle 57x^{4}-114x^{3}+118x^{2}-114x+57,$ (C.35)
$\displaystyle P_{31}$ $\displaystyle=$ $\displaystyle
57x^{4}+114x^{3}+110x^{2}+114x+57,$ (C.36) $\displaystyle P_{32}$
$\displaystyle=$ $\displaystyle 73x^{4}-192x^{3}+262x^{2}-192x+73,$ (C.37)
$\displaystyle P_{33}$ $\displaystyle=$ $\displaystyle
79x^{4}-214x^{3}+282x^{2}-214x+79,$ (C.38) $\displaystyle P_{34}$
$\displaystyle=$ $\displaystyle 99x^{4}-252x^{3}+310x^{2}-252x+99,$ (C.39)
$\displaystyle P_{35}$ $\displaystyle=$ $\displaystyle
111x^{4}-276x^{3}+338x^{2}-276x+111,$ (C.40) $\displaystyle P_{36}$
$\displaystyle=$ $\displaystyle 111x^{4}-200x^{3}+170x^{2}-200x+111,$ (C.41)
$\displaystyle P_{37}$ $\displaystyle=$ $\displaystyle
111x^{4}+60x^{3}+274x^{2}+60x+111,$ (C.42) $\displaystyle P_{38}$
$\displaystyle=$ $\displaystyle 373x^{4}+6435x^{3}-14420x^{2}+6435x+373,$
(C.43) $\displaystyle P_{39}$ $\displaystyle=$ $\displaystyle
28x^{5}-42x^{4}+35x^{3}+29x^{2}-117x+55,$ (C.44) $\displaystyle P_{40}$
$\displaystyle=$ $\displaystyle 70x^{5}-97x^{4}+13x^{3}+12x^{2}-2x+2,$ (C.45)
$\displaystyle P_{41}$ $\displaystyle=$ $\displaystyle
70x^{5}-11x^{4}+71x^{3}-70x^{2}-2x-2,$ (C.46) $\displaystyle P_{42}$
$\displaystyle=$ $\displaystyle x^{6}+10x^{5}-23x^{4}+40x^{3}-23x^{2}+10x+1,$
(C.47) $\displaystyle P_{43}$ $\displaystyle=$ $\displaystyle
x^{6}+12x^{5}+47x^{4}+96x^{3}+47x^{2}+12x+1,$ (C.48) $\displaystyle P_{44}$
$\displaystyle=$ $\displaystyle 4x^{6}+13x^{5}-38x^{4}+52x^{3}-38x^{2}+13x+4,$
(C.49) $\displaystyle P_{45}$ $\displaystyle=$ $\displaystyle
4x^{6}+21x^{5}+80x^{4}+60x^{3}+80x^{2}+21x+4,$ (C.50) $\displaystyle P_{46}$
$\displaystyle=$ $\displaystyle
27x^{6}-35x^{5}+85x^{4}-114x^{3}+85x^{2}-35x+27,$ (C.51) $\displaystyle
P_{47}$ $\displaystyle=$ $\displaystyle
27x^{6}-29x^{5}+15x^{4}-14x^{3}+15x^{2}-29x+27,$ (C.52) $\displaystyle P_{48}$
$\displaystyle=$ $\displaystyle
27x^{6}-28x^{5}-15x^{4}+38x^{3}-15x^{2}-28x+27,$ (C.53) $\displaystyle P_{49}$
$\displaystyle=$ $\displaystyle
27x^{6}+136x^{5}-245x^{4}-221x^{3}+202x^{2}+223x-134,$ (C.54) $\displaystyle
P_{50}$ $\displaystyle=$ $\displaystyle
36x^{6}-19x^{5}-67x^{4}-66x^{3}-67x^{2}-19x+36,$ (C.55) $\displaystyle P_{51}$
$\displaystyle=$ $\displaystyle 57x^{6}+4x^{5}-65x^{4}-16x^{3}-65x^{2}+4x+57,$
(C.56) $\displaystyle P_{52}$ $\displaystyle=$ $\displaystyle
57x^{6}+4x^{5}-57x^{4}-16x^{3}-57x^{2}+4x+57,$ (C.57) $\displaystyle P_{53}$
$\displaystyle=$ $\displaystyle
57x^{6}+224x^{5}+423x^{4}+344x^{3}+423x^{2}+224x+57,$ (C.58) $\displaystyle
P_{54}$ $\displaystyle=$ $\displaystyle
57x^{6}+224x^{5}+431x^{4}+216x^{3}+431x^{2}+224x+57,$ (C.59) $\displaystyle
P_{55}$ $\displaystyle=$ $\displaystyle
69x^{6}-905x^{5}-1667x^{4}+6264x^{3}-1856x^{2}-797x+24,$ (C.60) $\displaystyle
P_{56}$ $\displaystyle=$ $\displaystyle
75x^{6}-58x^{5}-2637x^{4}+5456x^{3}-2907x^{2}-598x-51,$ (C.61) $\displaystyle
P_{57}$ $\displaystyle=$ $\displaystyle
99x^{6}-212x^{5}+333x^{4}-448x^{3}+333x^{2}-212x+99,$ (C.62) $\displaystyle
P_{58}$ $\displaystyle=$ $\displaystyle
99x^{6}+68x^{5}+77x^{4}-96x^{3}+77x^{2}+68x+99,$ (C.63) $\displaystyle P_{59}$
$\displaystyle=$ $\displaystyle
111x^{6}+104x^{5}+217x^{4}-368x^{3}+217x^{2}+104x+111,$ (C.64) $\displaystyle
P_{60}$ $\displaystyle=$ $\displaystyle
285x^{6}-440x^{5}-4707x^{4}+10912x^{3}-6381x^{2}-872x-237,$ (C.65)
$\displaystyle P_{61}$ $\displaystyle=$ $\displaystyle
70x^{7}+15x^{6}-69x^{5}-94x^{4}+6x^{3}+127x^{2}-53x+2,$ (C.66) $\displaystyle
P_{62}$ $\displaystyle=$ $\displaystyle
27x^{8}-328x^{7}-1934x^{6}-2456x^{5}-7952x^{4}-2024x^{3}-962x^{2}+104x-27,$
(C.67) $\displaystyle P_{63}$ $\displaystyle=$ $\displaystyle
27x^{8}+494x^{7}+3382x^{6}+4750x^{5}+15904x^{4}+4210x^{3}+2410x^{2}-46x-27,$
(C.68) $\displaystyle P_{64}$ $\displaystyle=$ $\displaystyle
60x^{8}-537x^{7}-2164x^{6}+5223x^{5}+8561x^{4}+5277x^{3}-2002x^{2}-483x+33,$
(C.69) $\displaystyle P_{65}$ $\displaystyle=$ $\displaystyle
70x^{8}+173x^{7}-342x^{6}+245x^{5}+860x^{4}+241x^{3}-458x^{2}+45x-2,$ (C.70)
$\displaystyle P_{66}$ $\displaystyle=$ $\displaystyle
1280x^{8}+3209x^{7}+41288x^{6}+2239x^{5}-120320x^{4}+2239x^{3}+41288x^{2}+3209x$
(C.71) $\displaystyle+1280,$ $\displaystyle P_{67}$ $\displaystyle=$
$\displaystyle
17x^{10}+1828x^{9}-2261x^{8}-2392x^{7}-7422x^{6}+24304x^{5}-6162x^{4}-3688x^{3}$
(C.72) $\displaystyle-2531x^{2}+2476x-73,$ $\displaystyle P_{68}$
$\displaystyle=$ $\displaystyle
17x^{10}+2276x^{9}-2885x^{8}-24696x^{7}+91794x^{6}$ (C.73)
$\displaystyle-126000x^{5}+85278x^{4}-20808x^{3}-2867x^{2}+2060x-73,$
$\displaystyle P_{69}$ $\displaystyle=$ $\displaystyle
32x^{10}-1617x^{9}+1830x^{8}+1114x^{7}+1234x^{6}-9448x^{5}+5734x^{4}-506x^{3}$
(C.74) $\displaystyle+390x^{2}-807x-4,$ $\displaystyle P_{70}$
$\displaystyle=$ $\displaystyle
32x^{10}-1377x^{9}+82x^{8}+2442x^{7}-1938x^{6}-728x^{5}-1110x^{4}+1686x^{3}$
(C.75) $\displaystyle-134x^{2}-999x-4,$ $\displaystyle P_{71}$
$\displaystyle=$ $\displaystyle
37x^{10}-2748x^{9}+2823x^{8}+1256x^{7}+5114x^{6}-18896x^{5}+8822x^{4}-40x^{3}$
(C.76) $\displaystyle+1617x^{2}-2100x+19,$ $\displaystyle P_{72}$
$\displaystyle=$ $\displaystyle
37x^{10}-2592x^{9}-241x^{8}+4560x^{7}-2526x^{6}-1456x^{5}-3570x^{4}+3696x^{3}$
(C.77) $\displaystyle+137x^{2}-2160x+19,$ $\displaystyle P_{73}$
$\displaystyle=$ $\displaystyle
91x^{10}-3086x^{9}+5585x^{8}+19836x^{7}-87474x^{6}+126000x^{5}-89598x^{4}$
(C.78) $\displaystyle+25668x^{3}+167x^{2}-1250x-35,$ $\displaystyle P_{74}$
$\displaystyle=$ $\displaystyle
91x^{10}-2314x^{9}+2585x^{8}+3364x^{7}+5910x^{6}-24304x^{5}+7674x^{4}+2716x^{3}$
(C.79) $\displaystyle+2207x^{2}-1990x-35,$ $\displaystyle P_{75}$
$\displaystyle=$ $\displaystyle
208x^{10}+1947x^{9}-6722x^{8}+104x^{7}+33106x^{6}+61370x^{5}+33106x^{4}+104x^{3}$
(C.80) $\displaystyle-6722x^{2}+1947x+208,$ $\displaystyle P_{76}$
$\displaystyle=$ $\displaystyle
437x^{10}-866x^{9}-1849x^{8}+6412x^{7}-7694x^{6}+5232x^{5}-5930x^{4}+5764x^{3}$
(C.81) $\displaystyle-2407x^{2}-542x+419,$ $\displaystyle P_{77}$
$\displaystyle=$ $\displaystyle
437x^{10}-422x^{9}-3057x^{8}+148x^{7}+3786x^{6}-3312x^{5}$ (C.82)
$\displaystyle+4686x^{4}-68x^{3}-3327x^{2}-314x+419,$ $\displaystyle P_{78}$
$\displaystyle=$ $\displaystyle
557x^{10}-2012x^{9}+4515x^{8}+10256x^{7}-65160x^{6}+103636x^{5}-68040x^{4}$
(C.83) $\displaystyle+13496x^{3}+2715x^{2}-1472x+485,$ $\displaystyle P_{79}$
$\displaystyle=$ $\displaystyle 557x^{10}-1736x^{9}-5117x^{8}+18176x^{7}$
(C.84)
$\displaystyle-4312x^{6}-16556x^{5}-3304x^{4}+17528x^{3}-5333x^{2}-1412x+485,$
$\displaystyle P_{80}$ $\displaystyle=$ $\displaystyle
536x^{12}-2203x^{11}+807x^{10}-2747x^{9}+13604x^{8}-42x^{7}-16838x^{6}-42x^{5}$
(C.85) $\displaystyle+13604x^{4}-2747x^{3}+807x^{2}-2203x+536,$ $\displaystyle
P_{81}$ $\displaystyle=$ $\displaystyle
848x^{12}-1467x^{11}-3883x^{10}+10033x^{9}+11148x^{8}-4342x^{7}-21602x^{6}$
(C.86) $\displaystyle-4342x^{5}+11148x^{4}+10033x^{3}-3883x^{2}-1467x+848,$
$\displaystyle P_{82}$ $\displaystyle=$ $\displaystyle
2768x^{12}-13405x^{11}+1776x^{10}-116585x^{9}+235760x^{8}+331206x^{7}$ (C.87)
$\displaystyle-870752x^{6}+331206x^{5}+235760x^{4}-116585x^{3}+1776x^{2}-13405x+2768,$
$\displaystyle P_{83}$ $\displaystyle=$ $\displaystyle
2768x^{12}-10461x^{11}+15692x^{10}-6353x^{9}-33936x^{8}-3154x^{7}+83176x^{6}$
(C.88) $\displaystyle-3154x^{5}-33936x^{4}-6353x^{3}+15692x^{2}-10461x+2768.$

##  Appendix D Partially resummed expansions at $\hat{s}=4$

In the main text, we showed the partially resummed expansion of
$F^{(0)}_{V,1,1}(z)+\zeta_{2}F^{(0)}_{V,1,2}(z)+\zeta_{3}F^{(0)}_{V,1,3}(z)$.
We give now the corresponding results for the remaining factors. The
$G$–functions appearing in these expressions are given in Eqs. (4.53–4.56).

$\displaystyle
F^{(0)}_{V,2,1}(z)+\zeta_{2}F^{(0)}_{V,2,2}(z)+\zeta_{3}F^{(0)}_{V,2,3}(z)=$
(D.1)
$\displaystyle\frac{64\pi^{2}\frac{1}{\big{(}4-z^{2}\big{)}^{3/2}}G\left(\frac{1}{\tau},\sqrt{4-\tau}\sqrt{\tau};z^{2}\right)\ln(z)Q_{5}}{9z^{5}}+\frac{128\ln(2)\pi^{3}\frac{1}{\big{(}4-z^{2}\big{)}^{3/2}}\ln\big{(}1-\frac{z^{2}}{4}\big{)}Q_{14}}{27z^{5}}$
$\displaystyle+\frac{512\pi^{2}\ln^{2}(z)Q_{19}}{27(z-2)z^{4}(z+2)}-\frac{16\pi\frac{1}{\big{(}4-z^{2}\big{)}^{3/2}}\ln^{2}(z)Q_{60}}{27z^{3}}+\frac{2\ln(2)\pi^{3}\frac{1}{\sqrt{4-z^{2}}}Q_{63}}{243z^{5}}$
$\displaystyle-\frac{2\pi^{2}\ln(z)Q_{68}}{81(z-2)z^{2}(z+2)}+\ln(2)\pi^{3}\ln(z)\Biggl{[}-\frac{4\frac{1}{\sqrt{4-z^{2}}}Q_{3}}{27z^{5}}-\frac{4}{9}z^{3}\frac{1}{\big{(}4-z^{2}\big{)}^{3/2}}\Biggr{]}$
$\displaystyle+{\rm
Li}_{4}\left(\frac{1}{2}\right)\pi\Biggl{[}\frac{256}{3}z^{3}\frac{1}{\big{(}4-z^{2}\big{)}^{3/2}}+\frac{256\big{(}340+108z^{2}+27z^{4}\big{)}\frac{1}{\sqrt{4-z^{2}}}}{81z^{3}}$
$\displaystyle-\frac{2048\big{(}z^{2}-3\big{)}\frac{1}{\sqrt{4-z^{2}}}\ln\big{(}1-\frac{z^{2}}{4}\big{)}}{z^{5}}\Biggr{]}+\ln^{4}(2)\pi\Biggl{[}\frac{32}{9}z^{3}\frac{1}{\big{(}4-z^{2}\big{)}^{3/2}}$