Asymptotic behaviors of means of central values of automorphic L -functions for GL(2)GL(2)

Let AA be the adele ring of a totally real algebraic number field F  . We push forward an explicit computation of a relative trace formula for periods of automorphic forms along a split torus in GL(2)GL(2) from a square-free level case done by Masao Tsuzuki [20], to an arbitrary level case. By using a relative trace formula, we study central values of automorphic L  -functions for cuspidal automorphic representations of GL(2,A)GL(2,A) corresponding to Maass forms with arbitrary level.