Asymptotic formulas for the second moments of $L$-series associated to   holomorphic cusp forms on the critical line
  New uniform asymptotic formulas are obtained for the second moment of $L$-series of cusp forms of even weight $2k\ge2$ with respect to the congruence subgroup $\Gamma_0(N).$ 
V.A. Bykovskii, D.A. Frolenkov
math.NT