Dedekind zeta functions of quaternion number fields

There is the Brauer-Kuroda relation between the Dedekind zeta function
of a Galois extension F of Q and Dedekind zeta functions of some subfields of F.
If Gal(F/Q) is the group of quaternions, then the zeta function of F disappears
in the relation!
Nevertheless, after some numerical experiments we stated a conjecture relating
 the Dedekind zeta function of F with Artin L-functions. We can prove a particular
case of the conjecture applying representation theory of groups.