I know two statements about modular forms that are Riemann Hypothesis-ish.

First, note that the constant term of the level-one non-holomorphic Eisenstein series $E_s$ is $y^s+c(s)y^{-s}$, and that the poles of $c(s)$ are the same as the poles of $E_s$. We can directly calculate that $c(s)={\Lambda(s)\over\Lambda(1+s)}$ (this depends on your precise normalization of the Eisenstein series), where $\Lambda$ is the completed zeta function. We can actually say something about the location of the poles of $E_s$ (using the spectral theory of automorphic forms). Unfortunately, we only know how to control poles for ${\rm Re}(s)\ge 0$. This does give an alternate proof of the nonvanishing of $\zeta(s)$ at the edge of the critical strip (from the lack of poles of ${\Lambda(it)\over\Lambda(1+it)}$), but it doesn't seem possible to go further to the left (though it does generalize to other $L$-functions appearing as the constant term of cuspidal-data Eisenstein series).

Second, the values of modular forms at certain (Heegner) points in the upper-half plane can be related to zeta functions. For example, $E_s(i)={\Lambda_{{\mathbb Q}(i)}(s)\over \Lambda_{\mathbb Q}(2s)}$. The general statement is simple to express adelically. Take a quadratic extension $k_1$ of $k$, and let $H$ denote $k_1^\times$ as a $k$-group and $E_s$ the standard level-one Eisenstein series on $G=GL_2(k)$. Take a character $\chi$ on $Z_{\mathbb A}H_k\backslash H_{\mathbb A}$ then
$$\int_{Z_{\mathbb A}H_k\backslash H_{\mathbb A}}E_s(h)\chi(h)\ dh={\Lambda_{k_1}(s,\chi)\over \Lambda_k(2s)}$$

where $Z$ denotes the center of $G$, and we have normalized the measure on the quotient space to be 1. Note that since $H$ is a non-split torus in $G$, the quotient is compact, so the integral is finite. In fact, the integrand is invariant (on the right) under a compact open subgroup $K$ of $H_{\mathbb A}$, so the integral is actually over the double coset space $Z_{\mathbb A}H_k\backslash H_{\mathbb A}/K$, which is actually a finite group.

In order to get the Riemann zeta function in the numerator on the right-hand-side, you would need to integrate over a split torus, which is precisely the Mellin transform, and you would have convergence issues. Note that if it did converge, the Mellin transform of $E_s$ would be
$$\int_{Z_{\mathbb A}M_k\backslash M_{\mathbb A}} E_s(a)|a|^v\ da={\Lambda(v+s)\Lambda(v+1-s)\over\Lambda(2s)}$$

The second idea is more commonly discussed in the context of subconvexity problems for general $L$-functions. (See Iwaniec's Spectral Methods of Automorphic Forms, especially Chp 13.) A class of subconvexity results is the Lindelof Hypothesis, which is one of the stronger implications of the Riemann Hypothesis.