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.