Let E/Q be an elliptic curve of conductor N having ordinary reduction at a prime p and write f for the newform attached to it. Let K be a real quadratic field in which all the primes of N are split. We construct a p-adic analytic function L(k) which interpolates certain Shintani cycles attached to weight-k-specialization(s) of a Hida family interpolating f. When p divides N exactly, we show that L(k) vanishes to order at least 2 at k = 2, and express its second derivative at k = 2 as the product of the formal group logarithms of two global points on E defined over a quadratic extension of K. This may be regarded as an analogue for elliptic curves of a limit formula of Kronecker, in which Eisenstein series are replaced by cusp forms.