Extensive computer simulations have been carried out in order to obtain analytical expressions for such thermodynamic properties as the Helmholtz free energy, internal energy, entropy, equation of state, isothermal compressibility, thermal expansion coefficient, heat capacities at constant volume and constant pressure, etc., of a Lennard-Jones fluid. The proposed expressions combine accuracy with simplicity and they are valid over the entire supercritical region. The number of simulated states was 265, covering a range of temperatures T = (1.3-2.6) and densities from rho = (0.1-0.844) (both in reduced units). The given number of states permits us to fit the computer simulation data by the corresponding analytical expressions with a total guarantee of confidence. Because the heat capacity at constant pressure is related to the other quantities, such as the internal energy (through the heat capacity at constant volume) and to the equation of state (through the isothermal compressibility and the thermal expansion coefficient), in testing the goodness of our analytical relations we compared our C-p values with other theoretical values taken from the literature, and found excellent agreement. We also compared our C-p results with experimental data for argon, using our own recently proposed Lennard-Jones intermolecular parameters. Good agreement was again found.