Functional equation in the theory of zeta functions is one of the important components of this theory.
I am interested to know whether the similar property, having functional equation, for the spectral zeta functions has been studied? This might be a search for such a property of the spectral zeta function (for the Laplacian) of spaces with specific symmetries, e.g. symmetric spaces, or homogeneous ones.
The Selberg zeta function for, say compact Riemann surfaces (other results are known but I forget the exact generality), satisfies a functional equation. This is built out of eigenfunctions of the Laplacian, but perhaps is not what you mean by a "spectral zeta function."
If you mean something like the Minakshisundaram-Pleijel zeta function, then McKean, in his 1972 paper on Selberg's trace formula, that Minakshisundaram-Pleijel and Weil hoped this may have a functional equation, but there were no positive results thus far. A quick search reveals this paper from 1995 which obtains functional equations for the MP zeta function of spheres and real or complex projective spaces, but I didn't find any other cases where a functional equation has been established.
