Let $L(s)$ be an $L$-function coming from Hecke characters or automorphic forms (e.g. modular form on GL(2), Maass form on GL(2), and higher-rank analogues).
Is there any numerical evidence for Grand RH? For example, let $\tau$ be the Ramanujan tau function. Do we know that $L(s,\tau)$ is zero free (except critical line) under $\Im(s)< $ certain giant number? How about (transcendental) Maass forms on GL(2)?