to attack Theorem \[thm: main for split case\] is the following Whittaker function formula for $W_c$:

\[thm: whittaker function formula\] We have $W_c(1)\ne 0$ and $$W_c({{\mathrm{diag}}}(\varpi^m,1,\varpi^{-m}))=\frac{q^{-2m}}{t_1-t_1^{-1}}(t_1^{2m+1}-t_1^{-(2m+1)})W_c(1),$$ where $\varpi$ is a fixed uniformizer of $F_v$, $q$ is the number of the residue field of $F_v$ and $t_1=|\varpi|^\alpha$.

In fact, Theorem \[thm1.3\] is slightly general than the above Theorem \[thm: whittaker function formula\]. After Theorem \[thm: whittaker function formula\], a careful choice of test section $f_{s,v}\in I(s,\chi_v)$ (See §\[computation of the local zeta integral in a special case\] for the details) will give the desired property, i.e., $Z^*(W_c,f_{s,v})/L(s,\pi_v,{{\mathrm{Ad}}}\times \chi_v)$ has no zeros in the region ${{\mathrm{Re}}}(s)\ge 1/2$.

[After [@H18], Buttcane and Zhou [@BuZh Theorem 2.4] were able to show that the adjoint $L$-function of a Maass form for ${{\mathrm{SL}}}_3({\mathbb{Z}})$ must be entire. Their argument relies on a simple comparison between where the Gamma factor can have poles and where the Riemann zeta function can have zeros. Our result gives an extension of theirs to Maass forms for congruence subgroups of ${{\mathrm{SL}}}_3({\mathbb{Z}}).$]{}

[Theorem \[thm: main for split case\] shows that $L_f(s,\pi,{{\mathrm{Ad}}})$ has no poles in the region ${{\mathrm{Re}}}(s)\ge \frac{1}{2}$. To prove Conjecture \[conj: adjoint GL3 is entire\] in the case $n=3$, one still 