On the Effective Non-vanishing of Hecke--Maass $L$-functions at Special Points

In this paper, we consider the non-vanishing problem for the family of special Hecke--Maass $L$-values $L (1/2+it_f, f)$ with $f (z)$ in an orthonormal basis of (even or odd) Hecke--Maass cusp forms of Laplace eigenvalue $1/4 + t_f^2$ ($t_f > 0$). We prove that 33% of $L (1/2+it_f, f)$ for $t_f \leqslant T$ do not vanish as $T \rightarrow \infty$. For comparison, it is known that the non-vanishing proportion is at least 25% for the central $L$-values $L (1/2, f)$. Further, 33% may be raised to 50% conditionally on the generalized Riemann hypothesis. Moreover, we prove non-vanishing results on short intervals $|t_f-T| \leqslant T^{\mu}$ for any $0 < \mu < 1$. However, it is a curious case that the Riemann hypothesis does not yield better result for small $0 < \mu \leqslant 1/2$.