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added the following story to the Properties-section of Dedekind eta function and also to the Examples-section of functional determinant and zeta function of an elliptic differential operator:
For $E = \mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$ a complex torus (complex elliptic curve) equipped with its standard flat Riemannian metric, then the zeta function of the corresponding Laplace operator $\Delta$ is
$\zeta_{\Delta} = (2\pi)^{-2 s} E(s) \coloneqq (2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum} \frac{1}{{\vert k +\tau l\vert}^{2s}} \,.$The corresponding functional determinant is
$\exp( E^\prime(0) ) = (Im \tau)^2 {\vert \eta(\tau)\vert}^4 \,,$where $\eta$ is the Dedekind eta function.
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