Please note special day, time and location.   I will explain joint work with Kannan Soundararajan, where we find an "L-function analogue" of the Brun-Hooley sieve. Essentially, our method allows us to work analytically with long truncated Euler products inside the critical strip. As a consequence we obtain several new results on the distribution of the central values of families of L-functions. In particular I'll focus on consequences for the distribution of the Tate-Shafarevich group of (prime) twists of an elliptic curve.