case (when $H={{\mathrm{SL}}}_3$), among other things, Ginzburg proved that the above local zeta integral is Eulerian $$Z(\varphi,f_s)=\prod_v Z^*(W_v,f_{s,v}),$$ with $$Z^*(W_v,f_{s,v})=L(3s,\chi_v)L(6s-2,\chi_v^2)L(9s-3,\chi_v^3)\int_{N_2(F_v)\setminus H(F_v)}W_v(g)f_s(w_\beta g)dg,$$ where $\chi=\otimes_v\chi_v$ is a character of $F^\times\setminus {{\mathbb {A}}}^\times$ which is part of the datum of the Eisenstein series, $W_v$ is the Whittaker function associated with local component of $\varphi$, $N_2$ is certain subgroup of $H={{\mathrm{SL}}}_3$, and $w_\beta$ is the Weyl element associated with the long simple root $\beta$ of $G_2$. Ginzburg also showed that at unramified places, the local zeta integral $Z^*(W_v,f_{s,v})$ represents the adjoint $L$-function $L(s,\pi_v,{{\mathrm{Ad}}}\times \chi_v)$.

It is worth noting that both $Z(\varphi, f_s)$ depends only on the restriction of $\varphi$ to ${{\mathrm{SL}}}_3,$ and hence that one could, in principle begin with an automorphic representation of ${{\mathrm{SL}}}_3({\mathbb{A}})$ instead of ${{\mathrm{GL}}}_3({\mathbb{A}}).$ However, the unfolding argument in [@G] results in a particular Whittaker integral of $\varphi.$ Thus, one must restrict attention to representations which are globally generic with respect to this particular character. Now, each cuspidal automorphic representation of ${{\mathrm{SL}}}_3({\mathbb{A}})$ may be obtained as one of the components in the restriction of some cuspidal automorphic representation of ${{\mathrm{GL}}}_3({\mathbb{A}})$ (unique up to twist). As explained in [@BumpGinzburg p.120], the phenomenon that the integral $Z(\varphi,f_s)$ only depends on its 