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, 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 $N$ as well.

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.

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 $cT^{1/2}\log T\ll V\leq T$ (where $c>0$ is some large constant). See his paper: Nonvanishing of automorphic L-functions at special points, Acta Arith. 162 (2014), 309–335.

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, 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 $N$ as well.