  Observations of the redshifted 21-cm hyperfine line of neutral hydrogen from
early phases of the Universe such as Cosmic Dawn and the Epoch of Reionization
promise to open a new window onto the early formation of stars and galaxies. We
present the first upper limits on the power spectrum of redshifted 21-cm
brightness temperature fluctuations in the redshift range $z = 19.8 - 25.2$
($54-68$ MHz frequency range) using 14 hours of data obtained with the
LOFAR-Low Band Antenna (LBA) array. We also demonstrate the application of a
multiple pointing calibration technique to calibrate the LOFAR-LBA
dual-pointing observations centred on the North Celestial Pole and the radio
galaxy 3C220.3. We observe an unexplained excess of $\sim 30-50\%$ in Stokes
$I$ noise compared to Stokes $V$ for the two observed fields, which
decorrelates on $\gtrsim 12$ seconds and might have a physical origin. We show
that enforcing smoothness of gain errors along frequency direction during
calibration reduces the additional variance in Stokes $I$ compared Stokes $V$
introduced by the calibration on sub-band level. After subtraction of smooth
foregrounds, we achieve a $2\sigma$ upper limit on the 21-cm power spectrum of
$\Delta_{21}^2 < (14561\,\text{mK})^2$ at $k\sim 0.038\,h\,\text{cMpc}^{-1}$
and $\Delta_{21}^2 < (14886\,\text{mK})^2$ at $k\sim 0.038
\,h\,\text{cMpc}^{-1}$ for the 3C220 and NCP fields respectively and both upper
limits are consistent with each other. The upper limits for the two fields are
still dominated by systematics on most $k$ modes.
