We've been told over and over $\boxed{\zeta(-1) = 1 + 2 +3 + 4 + \dots = - \frac{1}{2}}$ can be do the same over number fields?
What should be the reasonable value for the zeta function $F = \mathbb{Q}(i) = \mathbb{Q}[x]/(x^2 + 1)$:
$$ \zeta_F(-1) = \sum_{ m + in \in \mathbb{Z}[i]} \sqrt{m^2 + n^2} = 0 + 4 \Big( 1 + \sqrt{2} + 2\sqrt{3} + 2 + 2\sqrt{5} +\dots \Big)$$
The Dedekind zeta function $\zeta_{\mathbb{Q}(i)}$ factors as a product between the usual $\zeta$ function and a Dirichlet $L$-function $L(s,\chi_4)$ associated with the Legendre symbol $\!\!\pmod{4}$. Given the reflection formula for the $\zeta$ function, it is enough to use the reflection formula for the previous $L$-function to find the zeta-regularization of your series. Why that number should be relevant, is another question.