Moments of L-functions and Liouville-Green method
  We show that the percentage of primitive forms of level one and weight $4k\rightarrow \infty$ 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. 
Olga Balkanova and Dmitry Frolenkov
math.NT math.CA