n} (-\lambda )^n \Bigr] \,\Delta(\lambda),  \\
\label{DPr}
\mbox{where}\quad \Delta(\lambda)  &\defi&  
\prod_{k=0}^\infty \Bigl( 1+{\lambda \over E_k} \Bigr)
\exp \Bigl[ \sum_{1 \le n \le \mu} {(-\lambda)^n \over n E_k^n} \Bigr] \quad
\mbox{(for all }\lambda), \\
\mbox{and}\quad {\tilde Z}(n) &=& Z(n) \quad \mbox{if $Z(s)$ is regular at $n$,
as when } n>\mu \\
{\tilde Z}(1) &=& \lim_{s \to 1}\Bigl( Z(s) - {c_{-1} \over s-1} \Bigr) \quad
(= Z(1) \mbox{ or its finite part})\end{aligned}$$ (the general formula for ${\tilde Z}(n)$ on a pole is more contrived and required only for $\mu \ge 2$ ([@V5], eq.(4.12)), whereas $\mu \le 3/2$ in this work). Lastly, by way of consequence, the determinants are also characterized by these Taylor series (converging for $|\lambda|<E_0$), $$\label{Dtay}
-\log D(\lambda) = Z'(0) + \sum_{n=1}^\infty {\tilde Z(n) \over n} (-\lambda)^n,
\quad  -\log \Delta(\lambda) = \sum_{n>\mu} {Z(n) \over n} (-\lambda)^n.$$ The simplest case is $\mu <1$: then $$\label{DFr}
\Delta(\lambda) = \prod_{k=0}^\infty \Bigl (1+{\lambda \over E_k} \Bigr) =
\exp \Bigl[ -\sum_{n=1}^\infty {Z(n) \over n} (-\lambda )^n \Bigr] , \quad
D(\lambda) \equiv \e^{-Z'(0)} \Delta(\lambda) .$$

Two other properties are worth mentioning:

– if a spectrum $\{ E_k\}$ is dilated to $\{\alpha E_k\}\ (\alpha>0)$, the spectral functions get rescaled to $$\label{Scal}
Z(s,\lambda \vert \alpha) \equiv \alpha^{-s} Z 