What does $L(n,\chi_4)$ mean? 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
$$
L(s,\chi) = \sum_{n=1}^{\infty}\frac{1}{n^s} = \zeta(s)
$$
