Functional equation of a special Dirichlet series
The author investigates the special Dirichlet series \[ L(s)=\frac{2}{\sqrt{3}}\sum^{\infty}_{n=1}\sin (\frac{2\pi n}{3})n^{-s},\quad s\in {\mathbb{C}}. \] This series converges uniformly in the half-plane \(Re(s)>1\) and thus represents an analytic function there. He shows that the function L can be extended to an analytic function in the whole complex plane. Since the function \[ G(t)=^{def}(e^ t+e^{-t}+1)^{-1} \] is analytic in \(| t| <\frac{2\pi}{3}\), it can be expanded into a power series in t there. Let (1) \(G(t)=\sum^{+\infty}_{n=0}a_ nt^{2n}\) in \(| t| <\frac{2\pi}{3}\). The main result in this paper is as follows: (1) L has zeros at -1,-3,-5,.... (ii) \(L(-2m)=(2m)!a_ m(m=0,1,2,...)\) where \(a_ m(m=0,1,2,...)\) are given by (1). \[ (iii)\quad L(2m+1)=(-1)^ m\frac{\sqrt{3}}{2}(\frac{2\pi}{3})^{2m+1}a_ m\quad (m=0,1,2,...). \]
scientific article; zbMATH DE number 4006598