Close coupling calculations of line shape parameters have been performed for the first pure rotational R0(j = 0–4) lines of CO in helium baths at various temperatures. Besides the usual Lorentzian widths and shifts, we provide the complex Dicke parameters as well as the double power law temperature representation of all four parameters. In addition, we study the speed dependence of these parameters. The R0(0) and R0(1) theoretical thermally averaged collisional widths and shifts between 500 and about 15 K are in excellent agreement with the values reported in the literature. Below this temperature range, we confirm the persistent substantial disagreement that exists since 1985 between experimental and theoretical values. We thus focus on this regime, which is important for astrophysical applications, and we discuss various beyond-Voigt effects at low temperatures to try to understand this mismatch. We show that such mechanisms do not allow experimental widths and shifts to be reconciled with those from theory.