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