I think this is widely open. Flicker has a conditional result under certain cases of the Artin conjecture for Artin $L$-functions, see the Theorem on Page 232 of Pacific J. Math. 154 (1992). In particular, his result shows that the adjoint square $L$-function is entire for $m=2,3,4$.

I think this is widely open. Flicker has a conditional result under certain cases of the Artin conjecture for Artin $L$-functions, see the Theorem on Page 232 of Pacific J. Math. 154 (1992). In particular, his result shows that the adjoint $L$-function is entire for $m=2,3,4$.

Added. As GFS remarked, the work of Ginzburg et al. shows, by the method of integral representations, that the adjoint $L$-function is entire for $m=3,4,5$.