This is (almost) rehashing what has been said but a with an intent to spell things out.

Using $r_2(k)=4\sum_{d\vert k}\left(\frac{-4}k\right)=4(1*\left(\frac{-4}k\right))(k)$, where $\left(\frac{a}b\right)$ is the Jacobi symbol,
and $\left(\frac{-4}{2k}\right)=0, \left(\frac{-4}{2k+1}\right)=(-1)^k$, 
while the Dirichlet series evolves under the arithmetic convolution, i.e., $\sum\frac{(a*b)(k)}{k^s}=\sum\frac{a(k)}{k^s}\sum\frac{b(k)}{k^s}$, it follows that
\begin{align}\sum_{k=1}^{\infty}\frac{r_2(k)}{k^s}
&=4\sum_{k=1}^{\infty}\frac{(1*\left(\frac{-4}k\right))(k)}{k^s}
=4\sum_{n=1}^{\infty}\frac1{n^s}\sum_{k=1}^{\infty}\frac{\left(\frac{-4}k\right)(k)}{k^s} \\
&=4\zeta(s)\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^s}=4\zeta(s)\beta(s).
\end{align}