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

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