  Formulas for the most general case of the zeta function associated to a
quadratic+linear+constant form (in {\bf Z}) are given. As examples, the
spectral zeta functions $\zeta_\alpha (s)$ corresponding to bosonic ($\alpha
=2$) and to fermionic ($\alpha =3$) quantum fields living on a noncommutative,
partially toroidal spacetime are investigated. Simple poles show up at $s=0$,
as well as in other places (simple or double, depending on the number of
compactified, noncompactified, and noncommutative dimensions of the spacetime).
This poses a challenge to the zeta-function regularization procedure.
