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 $2sigma$ upper limit on the 21-cm power spectrum of $Delta_{21}^2 < (14561,text{mK})^2$ at $ksim 0.038,h,text{cMpc}^{-1}$ and $Delta_{21}^2 < (14886,text{mK})^2$ at $ksim 0.038 ,h,text{cMpc}^{-1}$ 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.
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