For appropriate choices of $\theta^{\infty}_{1},\theta^{\infty}_{2}$, the $T^{3}$-periods of the holomorphic differentials $\tilde{\zeta}_{i}$ take values in $2\pi\sqrt{-1}\mathbb{Z}$. In particular, the holomorphic functions

$$Z_{i}=\exp(\int\tilde{\zeta}_{i}),\quad i=0,1,2$$

are defined without multivalue issues. For a suitable choice of multiplicative normalisation on $Z_{i}$, we have the functional equation

$$Z_{0}Z_{1}Z_{2}=1-e^{2\pi\sqrt{-1}\eta}.$$