Certainly, analytic continuation gives a unique, well-defined value at $0$, and generalizes Jacobi's formula:
$$ \frac{\mathrm{d} \det(A)}{\mathrm{d} \alpha} = \det(A) \operatorname{tr}\left(A^{-1} \frac{\mathrm{d} A}{\mathrm{d} \alpha}\right)$$
 from finite dimensional vector spaces, whenever analytic continuation is available.

If you take the circle with the Laplace operator, you actually will get the Riemann zeta function.

But the crucial difference between the above spectral zeta functions and the number theoretic is that the general zeta function *does not admit a functional equation*, in fact will not analytic continue to a meromorphic function of all of $\mathbb{C}$. they do analytically continue around a neighborhood of zero, if your eigenvalues do not grow to fast.

They do not grow to fast for pseudo differential operators on compact manifolds by a generalization of Weyl's law. In this case, there is a relation to index theory.