The answer is no. Consider G=Gl(n). We have $L(s,\pi)=L(s-s',\pi\otimes|det|^{s'})$. The absolute value is the adelic norm. In general,eg in Tate's thesis, the parameter s is sometimes avoided for this reason.
So your conjecture violates GRH. For n=1 and $\pi$ trivial over the rational numbers gives you a concrete counter example, because there exist zeros on the critical line.