In light of the previous application, we might want to try to find such a formula for:

The only possible singularities are at $0,z_0,-z_0$, since $\mathrm{\wp }$ is even, and $0$ is a pole of $\mathrm{\wp }(z)$ mod $L$. We can see that $z_0$ is removable by dividing the numerator and denominator by $z-z_0$ and taking the limit, or by noting that 1;;$z_0$ is a zero of order 1 in the denominator, and a zero of order at least one in the numerator.

We can further calculate the residues of $f$ at $0$ and at $-z_0$, and we get $-2$ and $2$ respectively. Applying the previous theorem, we get
$f(z)=2\zeta (z+z_0)-2\zeta (z)-2\zeta (z_0)$.