  Assuming the Riemann hypothesis, we prove the weak convergence of linear
statistics of the zeros of L-functions towards a Gaussian field, with
covariance structure corresponding to the $\HH^{1/2}$-norm of the test
functions. For this purpose, we obtain an approximate form of the explicit
formula, relying on Selberg's smoothed expression for $\zeta'/\zeta$ and the
Helffer-Sj\"ostrand functional calculus. Our main result is an analogue of the
strong Szeg{\H o} theorem, known for Toeplitz operators and random matrix
theory.
