Let $\pi$ be an automorphic representation on $\mathrm{GL}(m)/\mathbb{Q}$. Define $$L(s,\pi,\mathrm{Ad}):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an $L$-function with Euler product of degree $m^2-1$.

Is $L(s,\pi,\mathrm{Ad})$ holomorphic on the entire complex plane? As far as I know, this is known for $m=2$ by the work Shimura and Gelbart-Jacquet. What is know on the case of $m>2$?