Non-vanishing of Maass form L-functions at the critical point

In this paper, we consider the family ${L_j(s)}_{j=1}^{infty}$ of $L$-functions associated to an orthonormal basis ${u_j}_{j=1}^{infty}$ of even Hecke-Maass forms for the modular group $SL(2, Z)$ with eigenvalues ${lambda_j=kappa_{j}^{2}+1/4}_{j=1}^{infty}$. We prove the following effective non-vanishing result: At least $50 %$ of the central values $L_j(1/2)$ with $kappa_j leq T$ do not vanish as $Trightarrow infty$. Furthermore, we establish effective non-vanishing results in short intervals.