We present the second Open Gravitational-wave Catalog (2-OGC) of compact-binary coalescences, obtained from the complete set of public data from Advanced LIGOs first and second observing runs. For the first time we also search public data from the Virgo observatory. The sensitivity of our search benefits from updated methods of ranking candidate events including the effects of non-stationary detector noise and varying network sensitivity; in a separate targeted binary black hole merger search we also impose a prior distribution of binary component masses. We identify a population of 14 binary black hole merger events with probability of astrophysical origin $> 0.5$ as well as the binary neutron star merger GW170817. We confirm the previously reported events GW170121, GW170304, and GW170727 and also report GW151205, a new marginal binary black hole merger with a primary mass of $67^{+28}_{-17},mathrm{M}_{odot}$ that may have formed through hierarchical merger. We find no additional significant binary neutron star merger or neutron star--black hole merger events. To enable deeper follow-up as our understanding of the underlying populations evolves, we make available our comprehensive catalog of events, including the sub-threshold population of candidates and posterior samples from parameter inference of the 30 most significant binary black hole candidates.
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