  We compute in a direct (not algorithmic) way the zeta function of all
supersingular curves of genus 2 over a finite field k, with many geometric
automorphisms. We display these computations in an appendix where we select a
family of representatives of all these curves up to geometric isomorphism and
we exhibit equations and the zeta function of all their twists. As an
application we obtain a direct computation of the cryptographic exponent of the
Jacobians of these curves.
