In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups $\Gamma_0(N)^+$, where $N>1$ is a square-free integer. After we prove that $\Gamma_0(N)^+$ has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an "average" Weyl's law for the distribution of eigenvalues of Maass forms, from which we prove the "classical" Weyl's law as a special case. The groups corresponding to $N=5$ and $N=6$ have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for $\Gamma_0(5)^+$ than for $\Gamma_0(6)^+$. We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyl's laws. In addition, we employ Hejhal's algorithm, together with recently developed refinements from [31], and numerically determine the first $3557$ of $\Gamma_0(5)^+$ and the first $12474$ eigenvalues of $\Gamma_0(6)^+$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.

Abstract In this paper we obtain a full asymptotic expansion of the archimedean contribution to the Li coefficients λ F ( − n ) (n is a positive integer) attached to a function F in the certain class S ♯ ♭ of functions containing the Selberg class S and (unconditionally) the class of all automorphic L-functions attached to irreducible, unitary cuspidal representations of GL N ( Q ) . Applying the obtained results to automorphic L-functions, we improve the result of J.C. Lagarias concerning the asymptotic behavior of archimedean contribution to the nth Li coefficient attached to the automorphic L-function. We also deduce asymptotic behaviors of λ F ( − n ) , as n → + ∞ equivalent to Generalized Riemann Hypothesis (GRH) true and GRH false for F ∈ S ♯ ♭ .