(abridged) We present a statistical study of the low-mass X-ray binary (LMXB) populations of three nearby, old elliptical galaxies: NGC 3379, NGC 4278, and NGC 4697. With a cumulative ~1 Ms Chandra ACIS observing time, we detect 90-170 LMXBs within the D25 ellipse of each galaxy. Cross-correlating Chandra X-ray sources and HST optical sources, we identify 75 globular cluster (GC) LMXBs and 112 field LMXBs. In the low luminosity range allowed by our deeper data (LX < 5 x 1037 erg s-1), we find a significant relative lack of GC-LMXBs, when compared with field sources. Using the co-added sample from the three galaxies, we find that the incompleteness-corrected X-ray luminosity functions (XLFs) of GC and field LMXBs differ at ~4# significance at LX < 5 x 1037 erg s-1. As previously reported, these XLFs are consistent at higher luminosities. Our observations may indicate a potential predominance of GC-LMXBs with donors evolved beyond the main sequence, when compared to current models, but their efficient formation requires relatively high initial binary fractions in clusters. The field LMXB XLF can be fitted with either a single power-law model plus a localized excess at a luminosity of 5-6 x 1037 erg s-1, or a broken power-law with a similar low-luminosity break. This XLF may be explained with NS-red-giant LMXBs, contributing to ~15% of total LMXBs population at ~5x1037 erg s-1. The difference in the GC and field XLFs is consistent with different origins and/or evolutionary paths between the two LMXB populations, although a fraction of the field sources are likely to have originated in GCs.
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