  We prove effective versions of Oppenheim's conjecture for generic
inhomogeneous forms in the S-arithmetic setting. We prove an effective result
for fixed rational shifts and generic forms and we also prove a result where
both the quadratic form and the shift are allowed to vary. In order to do so,
we prove analogues of Rogers' moment formulae for $S$-arithmetic congruence
quotients as well as for the space of affine lattices. We believe the latter
results to be of independent interest.
