Is there any way to regularize this infinite product?

$$ \det \Delta = \prod_{a,b > 0} (a^2 + b^2) $$

I tried to say this was the derivative at $0$ of an L-function

$$ \zeta_{K}'(0) = \sum_{(a,b)} \log (a^2 + b^2) $$

and perhaps this can be evaluated using a functional equation (which I don't know off-hand).

I also said that on the torus $S^1 \times S^1$ as eigenvalues $a^2 + b^2$ on the basis (of characters) $e(a^2 + b^2) = e^{2\pi i(a^2 + b^2)}$.

$$ \left( \frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}\right) e^{2\pi i \, (ax + by)} = (a^2 + b^2 )e^{2\pi i \, (ax + by)} $$

Then I said we could write the Laplaian operator in the Fourrier basis:

$$ \Delta = \sum_{(a,b) \in \mathbb{Z}^2} \big( a^2 + b^2 \big) e^{2\pi i \, (ax + by)} $$

where hopefully this could make sense using theory of distributions.

This was motivated by trying to understand the Gross-Zagier formula, but I wound up using none of it.

• https://ncatlab.org/nlab/show/Dedekind+zeta+function
• https://en.wikipedia.org/wiki/Heegner_point

Also I found it odd to see the same regularization formulas over and over in physics papers, but sometimes, we can regulaize an L-function but there is no differential operator to go with it.