Nonvanishing of central L-values of Maass forms

For the full modular group $\mathrm{SL}_2(\mathbb{Z})$, Zhao Xu proved that a positive proportion of these $L$-values do not vanish, even when $\lambda_j$ is restricted to a short interval $\lambda_j\in[T-V,T+V]$ with $cT^{1/2}\log T\leq V\leq T$ (where $c>0$ is some large constant). See his paper: Nonvanishing of automorphic L-functions at special points, Acta Arith. 162 (2014), 309–335.

Added. This implies a positive proportion of nonvanishing for congruence subgroups $\Gamma_0(N)$ as well (assuming the nebentypus is trivial), because the spectrum of these include the spectrum of $\mathrm{SL}_2(\mathbb{Z})$ via oldforms. Probably Zhao Xu's proof can be extended to newforms of level $N$ (and any nebentypus) as well.

Apologies, I know this is a very late response to an old question. However, note that the paper referenced in the other answer shows that a positive proportion of the values $L(\frac12 + it_j, u_j)$ are nonvanishing in short intervals, where $u_j$ is a Maass cusp form with Laplace eigenvalue $\frac14 + t_j^2$, which has the "special point" $\frac12 + it_j$. I think the original question is about the central values $L(\frac12, u_j)$.

For $SL(2, \mathbb{Z})$, in Jeff Vanderkam's thesis he shows that at least $\frac16$ of $L(\frac12, u_j)$ are nonvanishing within a long interval; of course, $L(\frac12, u_j) = 0$ for odd $u_j$, so when restricting to even forms then the proportion is at least $\frac13$.

Shenhui Liu shows that a positive proportion of the central values are nonvanishing in short intervals–see his thesis or preprint https://arxiv.org/pdf/1702.07084.pdf.

The current best is due to Balkanova, Huang, and Södergren, who show that the proportion of nonvanishing at the central point (again, for even $u_j$) is at least $\frac12$–see their preprint https://arxiv.org/pdf/1810.07991.pdf.