A commutative noetherian local ring of embedding dimension 4 having a transcendental series of Betti numbers

We construct a ring with the properties of the title of the paper. We also construct some other local rings of embedding dimension 4 with exotic properties. Among the methods used are the {\tt Macaulay2}-package {\tt DGAlgebras} by Frank Moore, combined with and inspired by results by Anick, Avramov, Backelin, Katth"an, Lemaire, Levin, L"ofwall and others.