Central critical values of modular Hecke L-functions

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We give an explicit formula for the central critical value L(1/2,π̂⊗χ) of the base-change lift π̂ to an imaginary quadratic field K of an automorphic representation π as the square of a finite sum of the values of a nearly holomorphic cusp form in π at elliptic curves with complex multiplication by K. As long as the transcendental factor of the value is a CM period, χ is basically any unitary arithmetic Hecke character of K inducing the inverse of the central character of π.