  We investigate the response of 3D Luttinger semimetals to localized charge
and spin impurities as a function of doping. The strong spin-orbit coupling of
these materials strongly influences the Friedel oscillations and RKKY
interactions. This can be seen at short distances with an $1/r^4$ divergence of
the responses, and anisotropic behavior. Certain of the spin-orbital signatures
are robust to temperature, even if the charge and spin oscillations are smeared
out, and give an unusual diamagnetic Pauli susceptibility. We compare our
results to the experimental literature on the bismuth-based half-Heuslers such
as YPtBi and on the pyrochlore iridate Pr$_2$Ir$_2$O$_7$.
