Project B5 aims to extract the so-called L-invariant from special values of automorphic L-functions. This fine invariant appears naturally in the context of the p-adic Birch and Swinnerton-Dyer Conjecture: the order of vanishing of the p-adic L-function may be strictly larger than the order of vanishing of the complex L-function at the central critical point. Understanding the so-called exceptional zeroes is a crucial step towards the conjecture of Birch and Swinnerton-Dyer.