Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial \, \Omega $, i.e. acceptable parameters in the Dirichlet problem) and associated orthonormal basis (normalized eigenfunctions) $ \{ e_i(x)\}_{i=1}^\infty $, $x \in \Omega $. Then define the function $ \phi(s,x) = \sum_{n=1}^\infty \lambda_n^{-s} e_n(x) $ for $ s \in \mathbb{C} , x \in \Omega $. This series doesn't converge for $ \mathrm{Re} (s) \le l(x) \in \mathbb{R}$, but it can be analytically continued to all of $\mathbb{C}$ minus singularities (I don't recall my source for this fact, but do remember that I read it somewhere.) So I was wondering, if we use $ \langle g , h \rangle_{\Omega} = \int_{\Omega} f(x) \overline{g(x)} d\mu(x) $ as the natural inner product, and define $ \zeta_{\Omega}(s) = \sum \lambda^{-s} $ wherever it converges and analytically continued otherwise, is the following necessarily true? $ \langle \phi(\alpha,\cdot) , \phi(\beta,\cdot) \rangle_{\Omega} = \zeta_{\Omega}(\alpha + \beta^*).$
EDIT: It might help to reconsider this in a more general setting. Holding $\beta$ fixed above, the inner product can be seen as a functional $B:L^2(\Omega)\to\mathbb{C}$ acting on $\phi(\alpha,x)$ with respect to $x$. Similarly, analytic continuation may be seen as an operator $A:\Omega\to\Gamma\supset\Omega$ which, for $f\in C^\infty(\Gamma)$, sends the restriction $f|_\Omega$ to $f$ - in this case acting on $\phi(\alpha,x)$ with respect to $\alpha$. The question above is therefore a special case of a more generalized query: do $A$ and $B$ commute wherever their composition makes sense? If I had more background in category theory, I might express this in a morphism-flavored way lending itself to some elementary identity or other I'm not aware of, but I don't.