For appropriate choices of constants $0\leq\theta_{31}^{\infty},\theta_{32}^{\infty}\leq 2\pi$, the $T^{3}$-periods of the holomorphic differentials

(4.28)

$$\begin{cases}d\log z_{3}=\zeta_{3}-\sqrt{-1}(\theta_{13}^{\infty}d\eta_{1}+\theta_{23}^{\infty}d\eta_{2})\\ d\log z_{4}=\zeta_{4}+\sqrt{-1}(\theta_{13}^{\infty}d\eta_{1}+\theta_{23}^{\infty}d\eta_{2})-(K_{1}(a)d\eta_{1}-K_{2}(a)d\eta_{2})\end{cases}$$

take values in $2\pi\sqrt{-1}\mathbb{Z}$; here $K_{p}(a)$ are the constants defined in Lemma 4.25. In particular, the holomorphic functions $z_{3}$ and $z_{4}$ are defined without multivalue issues. For a suitable choice of multiplicative normalisation on $z_{3},z_{4}$ we have the functional equation

(4.29)

$$z_{3}z_{4}=f_{S}=1-z_{1}-z_{2}=1-e^{2\pi i\eta_{1}}-e^{2\pi i\eta_{2}}.$$