  We study the effect of resonances near the threshold of low energy
($\varepsilon$) reactive scattering processes, and find an anomalous behavior
of the $s$-wave cross sections. For reaction and inelastic processes, the cross
section exhibits the energy dependence $\sigma\sim\varepsilon^{-3/2}$ instead
of the standard Wigner's law threshold behavior $\sigma\sim\varepsilon^{-1/2}$.
Wigner's law is still valid as $\varepsilon\rightarrow 0$, but in a narrow
range of energies. We illustrate these effects with two reactive systems, a
low-reactive system (H$_2$ + Cl) and a more reactive one (H$_2$ + F). We
provide analytical expressions, and explain this anomalous behavior using the
properties of the Jost functions. We also discuss the implication of the
reaction rate coefficients behaving as $K\sim 1/T$ at low temperatures, instead
of the expected constant rate of the Wigner regime in ultracold physics and
chemistry.
