The second moment of the central values of the symmetric square L-functions

Abstract

Improving on a result by Luo, we prove that for $K^{\frac{1}{3}}\le G\le K$,

$$\begin{aligned} \sum _{\begin{array}{c} k\text { even}\\ K\le k\le K+G \end{array}}\sum _{f\in H_{k}}L^{2}\left( \frac{1}{2},\text {sym}^{2}f\right) \ll _{\epsilon }K^{1+\epsilon }G, \end{aligned}$$

where $H_{k}$ is a Hecke basis of holomorphic cusp forms of weight $k$.