Improvement on the fundamental plane of accreting black holes. Top panel:5 residuals between the original FP fitting equation of Du et al. (2016a) and the data (black circles). There is a significant linear trend; the black line traces the unweighted least squares best fit. The red circles show the residuals with the modified FP (see Sect. 5.3); there is no trend, and the best fitting (red) line is consistent with the zero slope (black line). Bottom panel: data points with L/L_Edd estimated with the original (black) and revised (red) FP equation.