Line shape parameters of the first pure rotational R(j = 0 − 4) lines of carbon monoxide in hydrogen baths, between 1 and 500 K, have been calculated using the close coupling method. The theoretical thermally averaged collisional widths and shifts between 30 and 500 K agree well with the values reported in the literature. However, below this temperature range, we confirm the long-standing substantial disagreement between experimental and theoretical values for the R(0) and R(1) lines. In addition to the usual collisional widths and shifts, we provide the complex optical frequency of the velocity-changing collisions. We also study the speed dependence of the line shape parameters and investigate their double power law temperature representation. We conclude that beyond-Voigt effects, including the collision duration, cannot reconcile theory and experiments at low temperatures.