  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 $T\rightarrow \infty$.
Furthermore, we establish effective non-vanishing results in short intervals.
