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Observations of the redshifted 21-cm hyperfine line of neutral hydrogen fromearly phases of the Universe such as Cosmic Dawn and the Epoch of Reionizationpromise to open a new window onto the early formation of stars and galaxies. Wepresent the first upper limits on the power spectrum of redshifted 21-cmbrightness temperature fluctuations in the redshift range $z = 19.8 - 25.2$($54-68$ MHz frequency range) using 14 hours of data obtained with theLOFAR-Low Band Antenna (LBA) array. We also demonstrate the application of amultiple pointing calibration technique to calibrate the LOFAR-LBAdual-pointing observations centred on the North Celestial Pole and the radiogalaxy 3C220.3. We observe an unexplained excess of $\sim 30-50\%$ in Stokes$I$ noise compared to Stokes $V$ for the two observed fields, whichdecorrelates on $\gtrsim 12$ seconds and might have a physical origin. We showthat enforcing smoothness of gain errors along frequency direction duringcalibration reduces the additional variance in Stokes $I$ compared Stokes $V$introduced by the calibration on sub-band level. After subtraction of smoothforegrounds, 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 upperlimits are consistent with each other. The upper limits for the two fields arestill dominated by systematics on most $k$ modes.

The first power spectrum limit on the 21-cm signal of neutral hydrogen during the Cosmic Dawn at z=20-25 from LOFAR