Moments of L-functions and Liouville-Green method

We show that the percentage of primitive forms of level one and weight $4k\to \mathrm{\infty }$ for which the associated $L$-function at the central point is no less than $(\log k{)}^{-2}$ is at least 20%. The key ingredients of our proof are the Kuznetsov convolution formula and the Liouville-Green method.