一类L-函数的函数方程

L-functionsand their functional equation s of the quadratic fields have played an important role inthe study of class nu mber and related problems. In ［2］ the authors studied theL-functions on a kind of real quadratic fields and obtained some interesting resu lts on akind of real quadratic fields with class number one. In this paper, as a furtherdevelopment of our study on such real quadratic fields, the authors ha ve proved thefollowing theorem.

Functional Equation for a Class of L-functions

L-functions and their functional equation s of the quadratic fields have played an important role in the study of class number and related problems. In [2] the authors studied the L-functions on a kind of real quadratic fields and obtained some interesting results on a kind of real quadratic fields with class number one. In this paper, as a further development of our study on such real quadratic fields, the authors have proved the following theorem.Theorem. Let d>1 be a fundamental discriminant and χ be a primitiverea l character mod k with k>1 and (d, k)=1. Other notations are the same as before. Then^L(s,χ,A) could be continued to be a holomorphic fun ction on the whole s-plane and satisfies the following functional equationφ(1-s)^L(1-s,x,A)=φ(s)^L(s,x,A),whereφ(s)=ds/2(k/π)sΓ((s+1)/2)2.