  We investigate the superfluidity of a three-dimensional weakly interacting
Bose gas with a one-dimensional Raman-type spin-orbit coupling at both zero and
finite temperatures. Using the imaginary-time Green's function within the
Bogoliubov approximation, we explicitly derive analytic expressions of the
current-current response functions in the plane-wave and zero-momentum phases,
from which we extract the superfluid density in the limits of long wavelength
and zero frequency. At zero temperature, we check that the resultant superfluid
density agrees exactly with our previous analytic prediction obtained from a
phase-twist approach. Both results also satisfy a generalized Josephson
relation in the presence of spin-orbit coupling. At finite temperature, we find
a significant non-monotonic temperature dependence of superfluid density near
the transition from the plane-wave phase to the zero-momentum phase. We show
that this non-trivial behavior might be understood from the sound velocity,
which has a similar temperature dependence. The non-monotonic temperature
dependence is also shared by Landau critical velocity, above which the
spin-orbit-coupled Bose gas loses its superfluidity. Our results would be
useful for further theoretical and experimental studies of superfluidity in
exotic spin-orbit coupled quantum gases.
