Let F be a totally real field, and let be a CM quadratic extension. We construct a p-adic L-function attached to Hida families for the group. It is characterized by an exact interpolation property for critical Rankin-Selberg L-values, at classical points corresponding to representations with the weights of smaller than the weights of. Our p-adic L-function agrees with previous results of Hida when splits above p or, and it is new otherwise. Exploring a method that should bear further fruits, we build it as a ratio of families of global and local Waldspurger zeta integrals, the latter constructed using the local Langlands correspondence in families. In an appendix of possibly independent recreational interest, we give a reality-TV-inspired proof of an identity concerning double factorials.
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