  We extend the calculus of adiabatic pseudo-differential operators to study
the adiabatic limit behavior of the eta and zeta functions of a differential
operator $\delta$, constructed from an elliptic family of operators indexed by
$S^1$. We show that the regularized values ${\eta}(\delta_t,0)$ and
$t{\zeta}(\delta_t,0)$ are smooth functions of $t$ at $t=0$, and we identify
their values at $t=0$ with the holonomy of the determinant bundle, respectively
with a residue trace. For invertible families of operators, the functions
${\eta}(\delta_t,s)$ and $t{\zeta}(\delta_t,s)$ are shown to extend smoothly to
$t=0$ for all values of $s$. After normalizing with a Gamma factor, the zeta
function satisfies in the adiabatic limit an identity reminiscent of the
Riemann zeta function, while the eta function converges to the volume of the
Bismut-Freed meromorphic family of connection 1-forms.
