  Ambient prime geodesic theorems provide an asymptotic count of closed
geodesics by their length and holonomy and imply effective equidistribution of
holonomy. We show that for a smoothed count of closed geodesics on compact
hyperbolic 3-manifolds, there is a persistent bias in the secondary term which
is controlled by the number of zero spectral parameters. In addition, we show
that a normalized, smoothed bias count is distributed according to a
probability distribution, which we explicate when all distinct, non-zero
spectral parameters are linearly independent. Finally, we construct an example
of dihedral forms which does not satisfy this linear independence condition.
