Hunting for some properties of a series similar to Riemann zeta function.

Question

I happened to meat a alternated zeta function in D. Borwein's paper(see eq. 40 of his paper). The series is $$ L_{-3}(s)=1-\frac{1}{2^s}+\frac{1}{4^s}-\frac{1}{5^s}+\frac{1}{7^s}-\frac{1}{8^s}+\dots $$ where the multiple of 3, i.e. 3,6,9,$\dots$ are neglected in the summation with the alternated sgn "+1" and "-1" unchanged. Since I am not really familiar with this kind of series or function, so I just wonder that is this kind of series has a name, or can you provide me with more information about the series?

Answer 1

A series of the form $$ \sum_{n=1}^{\infty} \frac{a_n}{n^s} $$ for some sequence $(a_n)_n$of complex numbers is called a Dirichlet series. When the sequence $(a_n)_n$ is of the form $a_n = \chi(n)$ for a Dirichlet character $\chi$, then it is called a Dirichlet $L$-Function.

This specific one is as mentioned in a commented one defined by a Dirichlet character modulo $3$, the non-prinicipal one.