Conformal field theories in 4 dimensions have a scale anomaly characterized by the coefficient c, which appears as the coefficient of logarithmic terms in momentum-space correlation functions of the energy-momentum tensor. By studying the CFT contribution to 4-graviton scattering amplitudes, we derive a sum rule for c in terms of OPE coefficients. The same momentum-space techniques can be used to obtain sum rules for other anomaly coefficients, such as the one appearing in correlation functions of conserved currents.