I was reading some articles related to Euler sums and the Riemann zeta function, when I came across this definition:
$$ L(n,\chi_4) = \sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^n} $$
What is this function called and how is it related to the zeta function?
That author is writing $\chi_4$ for a certain "character", defined by $$ \chi_4(n) = \begin{cases} 1, &n \equiv 1\pmod 4,\\ -1,&n\equiv 3\pmod 4,\\ 0, &\text{otherwise} \end{cases} $$ and then $$ L(s,\chi_4) = \sum_{n=1}^\infty \frac{\chi_4(n)}{n^s} = \sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^s} $$ is the corresponding "$L$-function" of that character.
The simple connection with the zeta function is that, taking a different character $\chi(n) = 1$ for all $n$, we get $$ L(s,\chi) = \sum_{n=1}^{\infty}\frac{1}{n^s} = \zeta(s) $$