Modified heat equations for an analytic continuation of the spectral $\zeta$ function

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.