Let \eta be [-11pc] [-7pc]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\gg 0. 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 stated by C. Deninger.