# Numerical Evidence for Grand Riemann Hypothesis?

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)?

• I think there have been numerical experiments with $\mathrm{GL}_2$ and $\mathrm{GL}_3$ cusp forms (Rubinstein, Farmer, Booker etc.). – GH from MO Mar 12 '17 at 22:35
• Maybe not "giant number," but GRH for $L(s,\tau)$ has been numerically tested. See H. Yoshida, "On calculations of zeros of $L$-functions related with Ramanujan's discriminant function on the critical line," J. Ramanujan Math. Soc. 3 (1988), no. 1, 87–95. (On MathSciNet it is MR0975839 (90b:11044)). – KConrad Mar 13 '17 at 3:31
• Lots of data at lmfdb.org – Felipe Voloch Mar 13 '17 at 3:32
• I thought the entire Rubinstein FRG (2008-12) was supposed to produce (literally) trillions of zeros of L-functions of all sorts of degrees. "We will test the Generalized Riemann Hypothesis for millions of L-functions of degrees 1,2,3,4 to large height: on the order of $10^{13}$ zeros will be computed for degree 1, $10^{10}$ zeros for degree 2, $10^8$ zeros for degree 3, and $10^7$ zeros for degree 4." What happened in the end? I'm not getting anything from his Waterloo website currently. If the LMFDB data has this, it is somewhat hidden. E.g., 11a has zeros up to height only 20? – post.as.a.guest Mar 13 '17 at 10:43