Central value of the symmetric square L-functions related to Hecke-Maass forms

Let {I center dot (j) (z) : j a parts per thousand yenaEuro parts per thousand 1} be an orthonormal basis of Hecke-Maass cusp forms with Laplace eigenvalue 1/4 + t (j) (2) . For each I center dot (j) (z), we have the automorphic L-function L(s, sym(2) I center dot (j) ), which is called the symmetric square L-function associated to I center dot (j) . In this paper, we consider the average estimate of L(1/2, sym(2) I center dot (j) ) and prove that, for sufficiently large T, the estimate holds for T (1/3 + epsilon) a parts per thousand currency signaEuro parts per thousand M a parts per thousand currency signaEuro parts per thousand T (1 -aEuro parts per thousand epsilon) .