L-shell X-ray production cross sections for 29Cu, 31Ga, 32Ge, 35Br, 39Y, 42Mo, 47Ag, 50Sn, 57La, 60Nd, 64Gd, 67Ho, 70Yb, 79Au and 82Pb were measured for carbon ions in the 2-12 MeV energy range. The copper to yttrium data were previously measured using a Si(Li) detector with a beryllium window, while the molybdenum to lead X-rays were counted with a windowless Si(Li) detector, whose efficiency was determined by various normalization techniques. The measured X-ray cross sections are compared to predicted X-ray cross sections from the first Born and the ECPSSR theories. Using single-hole fluorescence yields, at low projectile velocities the first Born approximation overpredicts the data by as much as one order of magnitude while the ECPSSR theory at worse underestimates the data by about a factor of four. For the highest projectile velocities relative to target L-shell electrons, both theories converge toward the data to almost within experimental uncertainties. After modifying the fluorescence yield to account for multiple outer-shell ionization, the ECPSSR theory is brought into better agreement with the data for light targets but still overestimates the data for heavier targets. Also, the 2 MeV measurements using doubly ionized carbon ions are still significantly underestimated by the ECPSSR theory even after accounting for multiple ionization. Our multiple ionization correction assumes that the target foil thickness is thin enough to insure single collision conditions. However, in order to obtain acceptable signal-to-noise ratios, the target foils used in this experiment were too thick to provide single collision conditions. For thicker foils, ions are stripped inside the target resulting in an appreciable fraction of higher charge states that enhance the ionization of outer shells and the effective fluorescence yields. This effect could account for the remaining discrepancy between the ECPSSR theory and the data.