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The visual brightness of the night sky is not a single-valued function of its brightness in other photometric bands, because the transformations between photometric systems depend on the spectral power distribution of the skyglow. We analyze the transformation between the night sky brightness in the Johnson-Cousins V band (mV, measured in magnitudes per square arcsecond, mpsas) and its visual luminance (L, in SI units cd m^-2) for observers with photopic and scotopic adaptation, in terms of the spectral power distribution of the incident light. We calculate the zero-point luminances for a set of skyglow spectra recorded at different places in the world, including strongly light-polluted locations and sites with nearly pristine natural dark skies. The photopic skyglow luminance corresponding to mV=0.00 mpsas is found to vary between 1.11-1.34 x 10^5 cd m^-2 if mV is reported in the absolute (AB) magnitude scale, and between 1.18-1.43 x 10^5 cd m^-2 if a Vega scale for mV is used instead. The photopic luminance for mV=22.0 mpsas is correspondingly comprised between 176 and 213 microcd m^-2 (AB), or 187 and 227 microcd m^-2 (Vega). These constants tend to decrease for increasing correlated color temperatures (CCT). The photopic zero-point luminances are generally higher than the ones expected for blackbody radiation of comparable CCT. The scotopic-to-photopic luminance ratio (S/P) for our spectral dataset varies from 0.8 to 2.5. Under scotopic adaptation the dependence of the zero-point luminances with the CCT, and their values relative to blackbody radiation, are reversed with respect to photopic ones.