We compute the distribution of the partition functions for a class of one-dimensional Random Energy Models (REM) with logarithmically correlated random potential, above and at the glass transition temperature. The random potential sequences represent various versions of the 1/f noise generated by sampling the two-dimensional Gaussian Free Field (2dGFF) along various planar curves. The method is based on an analytical continuation of the Selberg integral from positive integers to the complex plane. In particular, we unveil a duality relation satisfied by the suitable generating function of free energy cumulants in the high-temperature phase. It reinforces the freezing scenario hypothesis for that generating function, from which we derive the distribution of extrema for the 2dGFF on the [0,1] interval and unit circle. The results reported are obtained in collaboration with J.-P. Bouchaud, P. Le Doussal, and A. Rosso.