<s>[INST] $S(r,s) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+r+1)(2k+s+1)}$ for $r>s\geq0$ and even integers $S$ generates $\pi$ or rational values. \begin{align}
S(r,s) = \sum_{k=0}^{\infty} = \frac{(-1)^k}{(2k+r+1)(2k+s+1)}
\end{align}
Assuming $r>s\geq 0$ and both $r,s$ even integers.
A few values, using wolfram:
\begin{align}
&S(2,0) = \pi/4-1/2\\
&S(4,0) = 1/6\\
&S(4,2) =  5/6 - \pi/4\\
&S(6,0) = \pi/12-13/90\\
&S(6,2) = 1/30\\
&S(6,4) = \pi/4 - 23/30\\
&S(8,0) = 19/210\\
&S(8,2) = 181/630 - \pi/12
\end{align}
For the rational values of $S(r,s)$ we have:
\begin{align}
S(r,s) = \frac{1}{r-s} \left( \frac{1}{s+1}-\frac{1}{s+2}+\dots+\frac{1}{r-1} \tag{1} \right)
\end{align}
I think for the irrational values must be something similar to $(1)$ except we will have to add $\pi$ after the sum. 
What is the general formula for $S$ (or just the formula for irrational values) $?$ 
 [/INST] Write
$$H_z=\psi(z)+\gamma$$
where $\psi(z)$ is the digamma function. Then
$$H_{(z+1)/2}-H_{z/2}=2\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k+z}$$
$$H_{z+1}=\frac{1}{z}+H_z$$
for $z$ nonzero and not a negative integer (DLMF 5.7.7, DLMF 5.5.2). Additionally, if $0<p/q<1$ is a rational number in lowest terms, then
$$H_{p/q}=-\ln q-\frac{\pi}{2}\cot\left(\frac{\pi p}
{q}\right)+\frac{1}{2}\sum_{k=1}^{q-1}\cos\left(\frac{2\pi kp}{q}\right)\ln%
\left(2-2\cos\left(\frac{2\pi k}{q}\right)\right)$$
(DLMF 5.4.19).
You should be able to get a closed form for your $S(r,s)$ from there.
</s>