On the effective non-vanishing of Hecke–Maass L -functions at special points

In this article, we consider the non-vanishing problem for the family of special Hecke–Maass L -values L ( 1 / 2 + i t f , f ) L (1/2+itf, f) upper L left parenthesis 1 divided by 2 plus i t Subscript f Baseline comma f right parenthesis with f ( z ) $f (z)$ f left parenthesis z right parenthesis in an orthonormal basis of (even or odd) Hecke–Maass cusp forms of Laplace eigenvalue 1 / 4 + t f 2 1/4 + tf² 1 divided by 4 plus t Subscript f Superscript 2 ( t f > 0 tf> 0 t Subscript f Baseline greater than 0 ). We prove that 20% of L ( 1 / 2 + i t f , f ) L (1/2+itf, f) upper L left parenthesis 1 divided by 2 plus i t Subscript f Baseline comma f right parenthesis for t f ⩽ T tf T t Subscript f Baseline less than or slanted equals upper T do not vanish as T → ∞ T → ∞