Non-vanishing of Maass form L-functions at the central point.

In this paper, we consider the family {L_j(s)}_j=1^∞ of L-functions associated to an orthonormal basis {u_j}_j=1^∞ of even Hecke-Maass forms for the modular group SL(2,Z) with eigenvalues {λ_j = κ_j² + 1/4}_j=1^∞. We prove the following effective non-vanishing result: At least 50 % of the central values L_j(1/2) with κ_j ≤ T do not vanish as T → ∞. Furthermore, we establish effective non-vanishing results in short intervals. [ABSTRACT FROM AUTHOR]