Data of the correlation function, namely, the height covariance c1(r,t) for 1, 2 and 3 dimensions. It has been measured for the interface representation of a contact process system at its critical point. See arXiv:2304.10883 for more details. Each file contain two headers starting with #. The columns show, in order, the time, the coordinate r, the value of c1(r,t), the error estimate by the jackknife procedure and the numbers of files used. All the columns are separated by space. Data for one-dimensional system in: c1JK-1D-L8192-t4000000-f2000.dat Data for two-dimensional system in: c1JK-2D-L512-t200000-f498.dat Data for three-dimensional system in: c1JK-3D-L128-t40000-f20.dat The datasets have been produced and employed in the context of a scientific work currently published in arXiv:2304.10883.