We present a current catalog of 21 cm HI line sources extracted from the Arecibo Legacy Fast Arecibo L-band Feed Array (ALFALFA) survey over ~2800 square degrees of sky: the alpha.40 catalog. Covering 40% of the final survey area, the alpha.40 catalog contains 15855 sources in the regions 07h30m < R.A. < 16h30m, +04 deg < Dec. < +16 deg and +24 deg < Dec. < +28 deg and 22h < R.A. < 03h, +14 deg < Dec. < +16 deg and +24 deg < Dec. < +32 deg. Of those, 15041 are certainly extragalactic, yielding a source density of 5.3 galaxies per square degree, a factor of 29 improvement over the catalog extracted from the HI Parkes All Sky Survey. In addition to the source centroid positions, HI line flux densities, recessional velocities and line widths, the catalog includes the coordinates of the most probable optical counterpart of each HI line detection, and a separate compilation provides a crossmatch to identifications given in the photometric and spectroscopic catalogs associated with the Sloan Digital Sky Survey Data Release 7. Fewer than 2% of the extragalactic HI line sources cannot be identified with a feasible optical counterpart; some of those may be rare OH megamasers at 0.16 < z < 0.25. A detailed analysis is presented of the completeness, width dependent sensitivity function and bias inherent in the current alpha.40 catalog. The impact of survey selection, distance errors, current volume coverage and local large scale structure on the derivation of the HI mass function is assessed. While alpha.40 does not yet provide a completely representative sampling of cosmological volume, derivations of the HI mass function using future data releases from ALFALFA will further improve both statistical and systematic uncertainties.
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