Let $\zeta (z)$ be defined as before with respect to a lattice $L$ and a $\mathrm{\wp }$-function $\mathrm{\wp }(z)$. Then we can write:

This is just applying definitions:

We can use absolute convergence of the $\mathrm{\wp }$-function to switch the order of integration and actually integrate to get:

where the second part in the summand shows up naturally as the mittag leffler correction.

As a result of this, we have the following results:

1. $\zeta (z)$ is meromorphic with poles of order 1 on $L$, and with residues $1$. Therefore we also see that this function can't be elliptic, since 1;; poles must be of order at least 2
2. $\zeta$ is an odd function. We observe this by looking at the series and rearranging with absolute convergence.
3. By a similar trick we used to show $\mathrm{\wp }$ was 1;;elliptic, we have ${\eta }_1=\zeta (z+{\omega }_1)-\zeta (z)$ and ${\eta }_2=\zeta (z+{\omega }_2)-\zeta (z)$. Furthermore, ${\eta }_1=2\zeta \left(\frac{{\omega }_1}{2}\right)$ and ${\eta }_2=2\zeta \left(\frac{{\omega }_2}{2}\right)$
4. These coefficients satisfy a Legendere Relation: