Top panel: tSZ relativistic corrections to the tSZ spectral distortion as a function of the frequency for various temperatures of the hot plasma from 0 to 20 keV. Bottom panel: error on the tSZ spectral distortion induced by the linear approximation from 0 (dark blue solid line) to 10 keV (dark red solid line) as a function of the frequency, using T_e = 5 keV as a reference temperature.