  Let $\eta$ be a closed real 1-form on a closed Riemannian $n$-manifold
$(M,g)$. Let $d_z$, $\delta_z$ and $\Delta_z$ be the induced Witten's type
perturbations of the de~Rham derivative and coderivative and the Laplacian,
parametrized by $z=\mu+i\nu\in\mathbb C$ ($\mu,\nu\in\mathbb{R}$,
$i=\sqrt{-1}$). Let $\zeta(s,z)$ be the zeta function of $s\in\mathbb{C}$,
defined as the meromorphic extension of the function
$\zeta(s,z)=\operatorname{Str}({\eta\wedge}\,\delta_z\Delta_z^{-s})$ for $\Re
s\gg0$. We prove that $\zeta(s,z)$ is smooth at $s=1$ and establish a formula
for $\zeta(1,z)$ in terms of the associated heat semigroup. For a class of
Morse forms, $\zeta(1,z)$ converges to some $\mathbf{z}\in\mathbb{R}$ as
$\mu\to+\infty$, uniformly on $\nu$. We describe $\mathbf{z}$ in terms of the
instantons of an auxiliary Smale gradient-like vector field $X$ and the
Mathai-Quillen current on $TM$ defined by $g$. Any real 1-cohomology class has
a representative $\eta$ satisfying the hypothesis. If $n$ is even, we can
prescribe any real value for $\mathbf{z}$ by perturbing $g$, $\eta$ and $X$,
and achieve the same limit as $\mu\to-\infty$. This is used to define and
describe certain tempered distributions induced by $g$ and $\eta$. These
distributions appear in another publication as contributions from the preserved
leaves in a trace formula for simple foliated flows, giving a solution to a
problem of C.~Deninger.
