Panel a: available Doppler measurements for GJ 581 obtained with HIRES (green diamonds), HARPS (blue triangles) and CARMENES (red circles) mutually agree when fitted with a three-planet self-consistent N-body model. Panel b: time series plot for only the CARMENES data show that they are fully consistent with the best-fit three-planet dynamical model yielding an rms = 1.64 m s^-1, the lowest among the three data sets. Panel c: GLS power spectrum of the residuals from the three-planet dynamical model for GJ 581. The strong peak at 66.7 d was earlier attributed to a planet designated GJ 581 d, but is now believed to be induced by stellar activity. Several other peaks with periods near 81.8 d, 186.1 d and 371.2 d are significant, but unlikely of planetary nature and most likely also related to activity and the window function. Panel d: dynamical evolution of the best three-planet fit for the GJ 581 planetary system. This fit is stable for at least 10 Myr, but for illustrative purposes we show only a 2000 yr extent of the N-body integration, which clearly shows the perfectly synchronized orbital evolution. In this best-fit configuration the eccentricities of the three planets are oscillating with a period of ~500 yr in addition to the ~50 yr secular perturbations between GJ 581 e and GJ 581 b.