  Given a convex polytope, we define its geometric spectrum, a stacky version
of Batyrev's stringy E-functions, and we prove a stacky version of a formula of
Libgober and Wood about the E-polynomial of a smooth projective variety. As an
application, we get a closed formula for the variance of the geometric spectrum
and a Noether's formula for two dimensional Fano polytopes (polytopes whose
vertices are primitive lattice points). We also show that this geometric
spectrum is equal to the algebraic spectrum of the polytope (the spectrum at
infinity of a tame Laurent polynomial whose Newton polytope is the polytope
alluded to). This gives an explanation and some positive answers to Hertling's
conjecture about the variance of the spectrum of tame regular functions.
