Functional equation of a special Dirichlet series

Abstract

In this paper we study the special Dirichlet series L(s)=23∑n=1∞sin(2πn3)n−s, s∈C This series converges uniformly in the half-plane Re(s)>1 and thus represents a holomorphic function there. We show that the function L can be extended to a holomorphic function in the whole complex-plane. The values of the function L at the points 0,±1,−2,±3,−4,±5,… are obtained. The values at the positive integers 1,3,5,… are determined by means of a functional equation satisfied by L.

Keywords

• Dirichlet series
• analytic continuation
• functional equation
• Γ-function.