  We report on the direct cosmic relic neutrino background search from the
first two science runs of the KATRIN experiment in 2019. Beta-decay electrons
from a high-purity molecular tritium gas source are analyzed by a
high-resolution MAC-E filter around the kinematic endpoint at 18.57 keV. The
analysis is sensitive to a local relic neutrino overdensity of 9.7e10 (1.1e11)
at a 90% (95%) confidence level. A fit of the integrated electron spectrum over
a narrow interval around the kinematic endpoint accounting for relic neutrino
captures in the Tritium source reveals no significant overdensity. This work
improves the results obtained by the previous kinematic neutrino mass
experiments at Los Alamos and Troitsk. We furthermore update the projected
final sensitivity of the KATRIN experiment to <1e10 at 90% confidence level, by
relying on updated operational conditions.
