L-functions with Riemann’s functional equation and the Riemann hypothesis

Abstract

Let χ4 be the non-principal Dirichlet character mod 4 and $L(s,\chi_4)$ be the Dirichlet L-function associated with χ4 and put $R(s):= s 4^{s} L(s+1,\chi_4) + \pi L(s-1,\chi_4)$. In the present paper, we show that the function R(s) has the Riemann’s functional equation and its zeros only at the non-positive even integers and complex numbers with real part $1/2$. We also give other L-functions that have the same property.