Under RH, Montgomery has proven equidistribution results for the zeros of the Riemann Zeta function, which suggest a close connection of the distribution to certain results in Random matrix theory. Analogues have been proven for zeta function associated finite fields unconditionally.

The Riemann zeros with imaginary part less than $T$ grow like $T \log T$. The zeros of the Selberg zeta function of a compact surface, which are connected to the eigenvalues of the Laplace Beltrami operator, grow roughly like $T^2$, but here the Riemann hypothesis is true except for possible zeros with imaginary part being zero.

Nevertheless, I dare to ask, if there is something similar available for these eigenvalue?