  For an elliptic differential operator $D$ of order $h$ in $n$ dimensions, the
spectral $\zeta$-function $\zeta_D(s)$ for $\Re s > \frac{n}{h}$ can be
evaluated as an integral over the heat kernel $e^{-t D}$. Here, alternative
expressions for $\zeta_D(s)$ are presented involving an integral over kernels
$k_{n,m}$ for a modified heat equation, such that the integral is non-singular
around $s=0$, respectively close to potential poles around $s=\frac{m}{h},
m<n$. Besides explicit expressions for an analytic continuation of $\zeta_D(s)$
when $\Re s \leq \frac{n}{h}$, this provides an alternative method to study
functional determinants and the residues of $\zeta_D(s)$ that does not require
to compute Seeley-DeWitt coefficients explicitly to cancel divergences in the
heat trace.
