Let $\pi$ be an automorphic representation on GL($m$)/$\mathbb{Q}$.

Define $$L(s,\pi,Ad):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an L-function with euler products of degree $m^2-1$.

Is $L(s,\pi,Ad)$ a holomorphic L-function on the entire plane? As far as I know, this is known for m=2 from Shimura and Gelbart-Jacquet.

What's know on the case of $m>2$?