  We consider a class of singular Riemannian manifolds, the deformed spheres
$S^N_k$, defined as the classical spheres with a one parameter family $g[k]$ of
singular Riemannian structures, that reduces for $k=1$ to the classical metric.
After giving explicit formulas for the eigenvalues and eigenfunctions of the
metric Laplacian $\Delta_{S^N_k}$, we study the associated zeta functions
$\zeta(s,\Delta_{S^N_k})$. We introduce a general method to deal with some
classes of simple and double abstract zeta functions, generalizing the ones
appearing in $\zeta(s,\Delta_{S^N_k})$. An application of this method allows to
obtain the main zeta invariants for these zeta functions in all dimensions, and
in particular $\zeta(0,\Delta_{S^N_k})$ and $\zeta'(0,\Delta_{S^N_k})$. We give
explicit formulas for the zeta regularized determinant in the low dimensional
cases, $N=2,3$, thus generalizing a result of Dowker \cite{Dow1}, and we
compute the first coefficients in the expansion of these determinants in powers
of the deformation parameter $k$.
