Serre. In the mean time the fundamental contributions of Grothendieck and Deligne are explained: rationality and the functional equation for L-functions of l-adic sheaves in characteristic p, with essentially complete proofs; the theory of weights; and some theorems of Deligne on the Riemann hypothesis (the last of Weil’s conjectures), this time without proof. Notably, one will find an exposition of functional equations of the L-functions developed by Grothendieck, and a more precise statement and fairly complete parts of the proof of Grothendieck and Deligne’s theorem on the rationality and the functional equation of Hasse–Weil L-functions in characteristic > 0, confirming a conjecture of Serre in this case.