In the category of exponential motives over Q, there is a new extension of Q(-1) by Q(0) that does not come from classical motives. Its period matrix features Euler's constant, which one is tempted to think of as the regularised value of Riemann's zeta function at 1. I will discuss several results and open questions revolving around this extension, for example the role it plays as a "monodromy factor" for differential equations of E-functions. The talk is based on joint work with Peter Jossen.