${{\mathrm{Re}}}(s)\ge \frac{1}{2}.$ However, during the referee process for this paper, a gap in the proof of the main theorem in [@H18] was pointed out. In this paper, we first explain how to close this gap, and then extend [@H18 Theorem 6.1] to the finite part $L$-function $L_f(s,\pi,{{\mathrm{Ad}}}\times \chi)=L^{S_\infty}(s, \pi, {{\mathrm{Ad}}}\times \chi)$:

\[thm: main for split case\] Let $\chi$ be a unitary Hecke character of $F^\times\setminus {{\mathbb {A}}}^\times$ and $\pi$ be an irreducible cuspidal automorphic representation of ${{\mathrm{GL}}}_3({{\mathbb {A}}})$, then the finite part of the adjoint $L$-function $L_f(s,\pi, {{\mathrm{Ad}}}\times \chi)$ is holomorphic on the region ${{\mathrm{Re}}}(s)\ge 1/2$, except for a simple pole at ${{\mathrm{Re}}}(s)=1$ when $\chi$ is non-trivial and $\pi \cong \pi \otimes \chi$ $($which forces $\chi$ to be cubic$)$.

As mentioned above, to prove Theorem \[thm: main for split case\], we need to analyze Ginzburg’s local zeta integral at places $v\in S-S_\infty$, and compare the local zeta integral with the corresponding local L-function. More precisely, we need to show that for each place $v\in S-S_\infty$, there exists a “test" Whittaker function $W_v$ of $\pi_v$ and a “test" section $f_{s,v}\in I(s,\chi_v)$ such that the local zeta integral $Z^*(W_v,f_{s,v})$ “detects" all poles of $L(s,\pi_v,{{\mathrm{Ad}}}\times \chi_v)$ on the region ${{\mathrm{Re}}}(s)\ge 1/2$, i.e., the 