  We have studied the X-ray luminosity function (XLF) of low-mass X-ray
binaries (LMXBs) in the nearby lenticular galaxy NGC 3115, using the Megasecond
Chandra X-Ray Visionary Project Observation. With a total exposure time of ~1.1
Ms, we constructed the XLF down to a limiting luminosity of ~10^36 erg/s, much
deeper than typically reached for other early-type galaxies. We found
significant flattening of the overall LMXB XLF from dN/dL \propto
L^{-2.2\pm0.4} above 5.5x10^37 erg/s to dN/dL \propto L^{-1.0\pm0.1} below it,
though we could not rule out a fit with a higher break at ~1.6x10^38 erg/s. We
also found evidence that the XLF of LMXBs in globular clusters (GCs) is overall
flatter than that of field LMXBs. Thus our results for this galaxy do not
support the idea that all LMXBs are formed in GCs. The XLF of field LMXBs seems
to show spatial variation, with the XLF in the inner region of the galaxy being
flatter than that in the outer region, probably due to contamination of LMXBs
from undetected and/or disrupted GCs in the inner region. The XLF in the outer
region is probably the XLF of primordial field LMXBs, exhibiting dN/dL \propto
L^{-1.2\pm0.1} up to a break close to the Eddington limit of neutron star LMXBs
(~1.7x10^38 erg/s). The break of the GC LMXB XLF is lower, at ~1.1x10^37 erg/s.
We also confirm previous findings that the metal-rich/red GCs are more likely
to host LMXBs than the metal-poor/blue GCs, which is more significant for more
luminous LMXBs, and that more massive GCs are more likely to host LMXBs.
