  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.
