The Reimann Hypothesis (RH) for $L$-functions of modular forms states that all the non-trivial zeroes of an $L$-function of a modular form must lie on the critical line. My question is: why is this conjecture important? What concrete consequences would it have?
In terms of what I'm looking for: I'm clear about the consequences of the Reimann hypothesis for the ordinary Reimann zeta function. RH for the ordinary zeta function is is important because it tells us about the distribution of prime numbers and gives us an improved error term for the Prime Number Theorem.
But does the Reimann Hypothesis for modular forms give us any similarly concrete information? Or is it more of a standalone statement about $L$-functions that doesn't "mean" anything per se?
In my mind, I don't know why the location of the zeroes for $L$-functions of modular forms is important. All I know is that the "action" happens near $s=1$ because BSD links the behavior of the $L$-function to ranks of elliptic curves (at least for modular forms of weight $2$). I'm not sure, however, what meaning is attached to the zeroes of $L$-functions of modular forms in general (that is, away from $s=1$). Can we ascribe some meaning to the zeroes of these $L$-functions away from the point $s=1$ that makes RH important in this case?