We present the first sample of TDEs discovered during the SRG all-sky survey. These 13 events were selected among X-ray transients detected on the 0<l<180 hemisphere by eROSITA during its second scan of the sky (10 June-14 Dec. 2020) and confirmed as TDEs by our optical follow-up observations. The most distant event occurred at z=0.581. One TDE continued to brighten after its discovery for at least another 6 months. The X-ray spectra can be described by emission from a standard accretion disk with kT between 0.05 and 0.5 keV, consistent with near-critical accretion onto black holes of a few 10^3 to 10^8 Msun, although super-critical accretion is possibly taking place. In 2 TDEs, a spectral hardening is observed 6 months after the discovery, possibly indicating the formation of an accretion disk corona. 4 TDEs show an optical brightening concurring with or preceding the X-ray outburst. All 13 TDEs are optically faint, with Lg/Lx<0.1 in most cases, where Lg and Lx are the intrinsic g-band and 0.2-6 keV luminosities, respectively. This sample is thus drastically different from TDEs selected at optical wavelengths. We have constructed a TDE X-ray luminosity function in the 10^42.5-10^45 erg/s range. The TDE volume rate decreases with increasing X-ray luminosity approximately as a power law with alpha=-0.6+/-0.2. This is similar to a trend observed for optically selected TDEs. The total rate at z<0.6 is (1.1+/-0.5)10^-5 TDEs/galaxy/year, an order of magnitude lower than previously estimated from optical studies. This might indicate that X-ray bright events constitute a minority of all TDEs, which would provide support to models predicting a strong dependence on the viewing angle. Our current TDE detection threshold can be lowered by a factor of ~2, which should make it possible to find ~700 TDEs by the end of the SRG survey over the entire sky.
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