Chandra X-ray observations routinely resolve tens to hundreds of low-mass X-ray binaries (LMXBs) per galaxy in nearby massive early-type galaxies. These studies have raised important issues regarding the behavior of this population of remnants of the once massive stars in early-type galaxies, namely the connection between LMXBs and globular clusters (GCs) and the nature of the LMXB luminosity function (LF). In this paper, we combine five epochs of Chandra observations and one central field Hubble Space Telescope Advance Camera for Surveys observation of NGC 4697, one of the nearest, optically luminous elliptical (E6) galaxies, to probe the GC-LMXB connection and LMXB-LF down to a detection/completeness limit of (0.6/1.4) x 10^{37} ergs/s. We detect 158 sources, present their luminosities and hardness ratios, and associate 34 LMXBs with GCs. We confirm that GCs with higher encounter rates (Gamma_h) and redder colors (higher metallicity Z) are more likely to contain GCs, and find that the expected number of LMXBs per GC is proportional to Gamma_h^{0.79+0.18/-0.15} Z^{0.50+0.20/-0.18}, consistent with fainter X-ray sources in Galactic GCs and LMXBs in Virgo early-type galaxies. Approximately 11+/-2% / 8 +/-2% of GCs in NGC 4697 contain an LMXB at the detection/completeness limit. We propose that the larger proportion of metal-rich GCs in NGC 4697 compared to the Milky Way explains why these fractions are much higher than those of the Milky Way at similar luminosities. We confirm that a broken power-law is the best fit to the LMXB-LF, although we cannot rule out a cutoff power-law, and argue that this raises the possibility that there is no universal form for the LMXB-LF in early-type galaxies. We find marginal evidence for different LFs of LMXBs in GCs and the field and different spectra of GC-LMXBs and Field-LMXBs.
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