  Continuing our exploration of the collective properties of low-mass X-ray
binaries (LMXBs) in the stellar fields of normal galaxies, we compute in this
paper the expected X-ray luminosity function (XLF) of LMXBs, starting from the
results of the previous paper in the series (Paper II). We treat separately two
classes of LMXB evolution, the first being close systems whose initial orbital
periods are below the bifurcation period, wherein the companion is on the main
sequence when Roche-lobe contact occurs, the subsequent evolution is driven by
angular-momentum loss from the system, and the second being wider systems whose
initial orbital periods are above the bifurcation period, wherein the companion
is on the giant branch when Roche-lobe contact occurs, and the evolution of
these systems is driven by the nuclear evolution of the companion. We obtain
model luminosity profiles L(t) for individual LMXBs of both classes. We then
compute the LMXB XLF by folding in the inputs for the pre-LMXB collective
properties and formation rates with the above luminosity profiles. Because of
the long timescale on which LMXBs evolve, one needs to keep track of the
evolution of the star-formation rate (SFR) on the same timescale, and we use
star-formation histories given by canonical models. We compare the observed
LMXB XLF with our computed one, keeping in mind that we have included only
neutron-star systems in this work, so that there would be an unaccounted-for
population of black-hole binaries at the high-luminosity end. We show that a
qualitative similarity already exists between the two, and we discuss the role
of the giant fraction, i.e., that fraction of all LMXBs which harbors a
low-mass giant companion, on the shape of the XLF at the high-luminosity end.
