I will explain on a very concrete example what Beilinson’s conjecture predicts for the value at 2 of the L-function of an elliptic curve.

As a consequence, it should become clear that there is an `unexplained’ entry in the associated period matrix, for which no conjectural description presently exists. In the second half of the talk I will introduce mixed L-functions, and illustrate on examples how their values provide many interesting periods in mathematics, including the missing entries of the period matrix described in the first part of the talk.