Integral mean values of L-functions

Let f ∈ Sk (Gamma0(N)) be a cusp form of weight k and level N, and L(s, f) its normalized L-function. This paper is devoted to the investigation of the integral mean values for L( s, f) along the critical line Rs = 1/2, and obtains an asymptotic formula in the case that the level N is squarefree. A theory of nonholomorphic Poincare series for GL(2, R ) is developed. Further generalizations to non-squarefree levels and to Maass forms are also indicated.