| The theory of automorphic L-functions is one of the important topics in contemporary number theory,which is closely related to many number theory problems,such as Riemann Conjecture which is the Millennium Mathematical Puzzle,BSD Conjecture,and the existence of solutions to Diophantine equations.The problem of estimating the average values of automorphic L-functions is an important research direction for in-depth understanding of the properties of automorphic L-functions.Relevant conclusions also have very important applications in number theory,and have been widely concerned and highly valued by number theorists.In this paper,we study the following weighted average values of automorphic Lfunctions on the critical line (?)where S2*(q)denotes the set of primitive Hecke eigenforms of weight 2 and prime level q,λf(pj)is the pj-th normalized Fourier coefficient of f,L(s,f)=(?)λf(n)n-s and L(s,f)is the L-function associated to f.In this paper,we give the asymptotic formula for the second moment of the automorphic L-function in the case of t=0 and 0≠t∈R.At the same time,we further study the asymptotic formula for the second moment of the automorphic L-function when the weighted Fourier coefficient is λf(p2). |
