Real parts of non-trivial zeros of L-functions

Question

Let $L_{\pi}$ be the L-function associated to an automorphic representation $\pi$ of $\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$, and let $R_{L_{\pi}}$ denote $\{\Re(s)-\frac{1}{2}|L_{\pi}(s)=0\wedge 0\lt \Re(s)\lt 1\}$. Is this last set proven to be compact?

Answer 1

Before asking a question in the greatest possible generality, it is worth stopping to think what is known in the simplest possible cases. In this case, the answer is: not much. Indeed, compactness of this set in particular implies that it has a maximal element $1-c<1$. This implies that the function at hand has no zeros in the strip $1-c<\operatorname{Re}(s)<1$. Even for the Riemann zeta function this is a wide open problem, equivalent to (say) error terms in PNT way stronger than we have available.

I would dare to guess that there is no single automorphic L-function for which compactness is known.