Are there any results on the proportion of nonzero central L-values of Maass cusp forms? More precisely, I am looking for lower bounds for


\begin{equation} \frac{\#\{\phi\_j : \, L(1/2, \phi\_j) \neq 0, \, \lambda\_j \leq T\}}{\#\{\phi\_j : \, \lambda\_j \leq T\}} \end{equation}


as $T \rightarrow \infty$, where $\phi\_j$ (with eigenvalue $\lambda\_j$) is an orthonormal basis of Maass cusp forms for a congruence subgroup of $\mathrm{SL}\_2(\mathbb{Z})$.


I ask this question as I found results of this kind for L-functions of holomorphic cusp forms in Iwaniec-Sarnak's work ''The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros'' and for Rankin-Selberg L-functions of Maass forms and a fixed holomorphic cusp form in Luo's work ''Nonvanishing of L-Values and the Weyl Law''.



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.


