We study hydrogen in the Saha regime, within the physical picture in terms of a quantum proton-electron plasma. Long ago, Saha showed that, at sufficiently low densities and low temperatures, the system behaves almost as an ideal mixture made with hydrogen atoms in their groundstate, ionized protons and ionized electrons. More recently, that result has been rigorously proved in some scaling limit where both temperature and density vanish. In that Saha regime, we derive exact low-temperature expansions for the pressure and internal energy, where density rho is rescaled in units of a temperature-dependent density rho* which controls the cross-over between full ionization (rho << rho*) and full atomic recombination (rho >>rho*). Each term reduces to a function of rho/rho* times temperature-dependent functions which decay exponentially fast when temperature T vanishes. Scaled expansions are ordered with respect to the corresponding decay rates. Leading terms do reduce to ideal contributions obtained within Saha theory. We consistently compute all corrections which are exponentially smaller by a factor exp(beta E (H) ) at most, where E (H) is the negative groundstate energy of a hydrogen atom and beta=1/(k (B) T). They include all effects arising from both the Coulomb potential and the quantum nature of the particles: excitations of atoms H, formation of molecules H-2, ions H-2(+) and H-, thermal and pressure ionization, plasma polarization, screening, interactions between atoms and ionized charges, etc. Scaled low-temperature expansions can be viewed as partial resummations of usual virial expansions up to arbitrary high orders in the density.