A web of conjectures predicts that the values of L-functions of automorphic forms at integer points are closely related to properties of the Galois representations attached to these automorphic forms. The adjoint L-function, which exists for every automorphic form, is expected to play an especially important role: its value at s = 1 should measure the number of ways the Galois representation can vary in a continuous family. This principle was established by Diamond, Flach, and Guo for the L-functions of modular forms, and by Dimitrov for Hilbert modular forms; their work was based in part on ideas due to Hida and made important use of the Taylor-Wiles method. For more general automorphic forms, the analogue of Hida's insights has been lacking. I will explain how a recent conjecture of Ichino and Ikeda may provide a substitute for Hida's arguments.