series representation for them in terms of zeta function derivatives at integers $j \geq 2$.

For a collection of formulae involving series of zeta function values we mention Ref. [@sri].

We introduce the constants $$d_j(a) \equiv \int_0^\infty {{\ln^{j-1} (t+a)} \over {(t+a)}} \ln\left({{[t]+a+1}
\over {t+a+1}}\right )dt, ~~~~~~~~~~j \geq 1,\eqno(5)$$ and $$c_j \equiv \int_1^\infty {{\ln^{j-1} t} \over t} \ln\left({{[t]+1}
\over {t+1}}\right )dt=d_j(1), ~~~~~~~~~~j \geq 1.  \eqno(6)$$ The approximate values of the first few of the latter constants are $c_1 \simeq -0.334$, $c_2 \simeq -0.433$, and $c_3 \simeq -0.93$. We have in general from the definition (6) that $c_j < 0$. We then have [**Proposition 1**]{}. For integers $j \geq 1$ we have (a) $$\gamma_j = (-1)^j\sum_{m=2}^\infty {{(-1)^m} \over m} \zeta^{(j)}(m)
-(-1)^j j! (1-2^{-j})\zeta(j+1)- j c_j,  \eqno(7)$$ (b) $$\gamma_j(a) = (-1)^j\sum_{m=2}^\infty {{(-1)^m} \over m} \zeta^{(j)}(m,a)
-j\sum_{\ell=0}^{j-1} (-1)^\ell \ell !{{j-1} \choose \ell} \ln^{j-\ell-1} a ~\mbox{Li}_{\ell+2}(-a)$$ $$-jd_j(a)-\ln^{j+1} a, \eqno(8)$$ (c) for integers $n \geq 2$ $$S_\gamma(n)=1-\ln 2-(2 \ln 2+\gamma-1)n+\int_1^\infty {{P_1(x)} \over x}
\left[{1 \over x}-\ln\left(1+{1 \over x}\right)\right] L_{n-2}^2(\ln x)dx$$ $$-\int_1^\infty {{P_1(x)} \over {x^2(1+x)}}\left[n-L_{n-1}^1(\ln x)\right]dx
+\sum_{m=3}^\infty {{(-1)^m} \over m}\left({{m-2} \over {m-1}}\right)^n$$ $$+\int_0^\infty \left[L_{n-1}^1(-x)-L_{n-1}^1(-x/2)\right]{{dx}
\over {(e^x -1)}}+\int_1^\infty {{L_{n-2}^2(\ln t)} \over t}
\ln\left({{[t]+1} \over {t+1}}\right )dt,  \eqno(9)$$ and (d) with $f(t)=\ln(t+1)/t(t+1)$ $$\sum_{j=1}^\infty {{(-1)^{j-1}} \over {j!}}(jc_j)={\pi^2 \over {12}}
+{1 