Maass尖形式对应的对称L-函数的特殊值问题

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

设{φj(z):j≥1}是Hecke-Maass尖形式的一组正交基,对应的特征值为1/4+t2/j.令L(s,sym2φj)是由φj(z)提升得到的对称平方L-函数.对充分大的T,本文给出了如下渐进公式:∑je-tj2/T2αjL(1/2,sym2φj)=T2P(logT)+O(T3/2+ε),其中αj是一个固定的权函数,P(x)是次数为1的多项式.

Let {Фj（z）：j≥1） be an orthonormal basis of Hecke-Maass cusp forms with Laplace eigenvalue 1/4＋t2j. For each Фj（z）, we have the automorphic L-function L（s,sym2Фj） which is called the symmetric square L-function associated to Фj. In this paper, the average estimate of L（1/2, sym2Фj） is considered, i.e., for sufficiently large T, the asymptotic formula j∑e-t2j/T2αjL（1/2,sym2Фj）=T2P（logT）＋O（t2^3＋ε） is established, where αj is a certain fixed weight function and P（x） is a polynomial of degree 1.