Values of L-functions and periods

I will begin by illustrating on a simple example how the value
of the L-function of an elliptic curve at the point 2 may be interpreted
as a period integral associated to a simple extension, in accordance
with Beilinson’s conjecture. However, this extension involves an
additional period which has no known interpretation. In the second half
of the talk I will propose a notion of mixed L-function whose values
supply this missing period.