The first accurate and complete theoretical room-temperature rotationally resolved spectra in the range 300-3000 cm-1 are reported for the three most abundant isotopologues (32SF6, 33SF6 and 34SF6) of the sulfur hexafluoride molecule. The literature reports that SF6 is widely used as a prototype molecule for studying the multi-photon excitation processes with powerful lasers in the infrared range. On the other hand, SF6 is an important greenhouse molecule with a very long lifetime in the atmosphere. Because of relatively low vibrational frequencies, the hot bands of this molecule contribute significantly to the absorption infrared spectra even at room temperature. This makes the calculation of complete rovibrational line lists required for fully converged opacity modeling extremely demanding. In order to reduce the computational costs, symmetry was exploited at all stages of the first global variational nuclear motion calculations by means of irreducible tensor operators. More than 2600 new vibrational band centers were predicted using our empirically refined ab initio potential energy surface. Highly excited rotational states were calculated up to J = 121, resulting in 6 billion transitions computed from an ab initio dipole moment surface and distributed over more than 500 cold and hot bands. The final line lists are made available through the TheoReTS information system (http://theorets.univ-reims.fr, http://theorets.tsu.ru). For the first time, the major (ro)vibrational band structures in the wavenumber range corresponding to the strongest absorption in the infra-red are completely elucidated for a seven-atom molecule, providing excellent agreement with the observed spectral patterns. It is shown that the obtained results are more complete than all available line lists, permitting reliable modelling of the temperature dependence of the molecular opacity.