Special values of the spectral zeta functions for locally symmetric Riemannian manifolds

Abstract

In this paper, we establish the formulas expressing the special values of the spectral zeta function ξ_Δ(n) of the Laplacian Δ on some locally symmetric Riemannian manifold Γ\bsla G/K in terms of the coefficients of the Laurent expansion of the corresponding Selberg zeta function. As an application, we give a numerical estimation of the first eigenvalue of Δ by computing the values ξ_Δ(n) numerically, when Γ\bsla G/K is a Riemann surface with Γ being the quaternion group.