  The kinetic instabilities of the Weibel-type are presently invoked in a large
variety of astrophysical scenarios because anisotropic plasma structures are
ubiquitous in space. The Weibel instability is driven by a temperature
anisotropy which is commonly modeled by a bi-axis distribution function, such
as a bi-Maxwellian or a generalized bi-Kappa. Previous studies have been
limited to a bi-Kappa distribution and found a suppression of this instability
in the presence of suprathermal tails. In the present paper it is shown that
the Weibel growth rate is rather more sensitive to the shape of the anisotropic
distribution function. In order to illustrate the distinguishing properties of
this instability a \emph{product-bi-Kappa distribution} is introduced, with the
advantage that this distribution function enables the use of different values
of the spectral index in the two directions, $\kappa_{\parallel} \ne
\kappa_{\perp}$. The growth rates and the instability threshold are derived and
contrasted with those for a simple bi-Kappa and a bi-Maxwellian. Thus, while
the maximum growth rates reached at the saturation are found to be higher, the
threshold is drastically reduced making the anisotropic product-bi-Kappa (with
small kappas) highly susceptible to the Weibel instability. This effect could
also rise questions on the temperature or the temperature anisotropy that seems
to be not an exclusive source of free energy for this instability, and
definition of these notions for such Kappa distributions must probably be
reconsidered.
