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Context. The HARPS spectrograph provides state-of-the-art stellar radial velocity (RV) measurements with a precision down to 1 m/s. The spectra are extracted with a dedicated data-reduction software (DRS) and the RVs are computed by CCF with a numerical mask. Aims. The aim of this study is three-fold: (i) Create easy access to the public HARPS RV data set. (ii) Apply the new public SERVAL pipeline to the spectra, and produce a more precise RV data set. (iii) Check whether the precision of the RVs can be further improved by correcting for small nightly systematic effects. Methods. For each star observed with HARPS, we downloaded the publicly available spectra from the ESO archive and recomputed the RVs with SERVAL. We then computed nightly zero points (NZPs) by averaging the RVs of quiet stars. Results. Analysing the RVs of the most RV-quiet stars, whose RV scatter is < 5 m/s, we find that SERVAL RVs are on average more precise than DRS RVs by a few percent. We find three significant systematic effects, whose magnitude is independent of the software used for the RV derivation: (i) stochastic variations with a magnitude of 1 m/s; (ii) long-term variations, with a magnitude of 1 m/s and a typical timescale of a few weeks; and (iii) 20-30 NZPs significantly deviating by a few m/s. In addition, we find small (< 1 m/s) but significant intra-night drifts in DRS RVs before the 2015 intervention, and in SERVAL RVs after it. We confirm that the fibre exchange in 2015 caused a discontinuous RV jump, which strongly depends on the spectral type of the observed star: from 14 m/s for late F-type stars, to -3 m/s for M dwarfs. Conclusions. Our NZP-corrected SERVAL RVs can be retrieved from a user-friendly, public database. It provides more than 212 000 RVs for about 3000 stars along with many auxiliary information, NZP corrections, various activity indices, and DRS-CCF products.