  This article is the first one of a series of three articles devoted to
L-functions. In this one we give a definition of the L-functions of convergent
or overconvergent F-modules with the help of Teichm\"uller liftings and we
establish the meromorphy of the L-functions of convergent F-modules in the
closed unit disk. Wan has established Dwork conjecture in a series of three
articles; owing to an isogeny theorem of Katz, his proof reduces to the
ordinary case: here we prove, on two explicit examples related to families of
elliptic curves, that the slope filtration on an ordinary overconvergent
F-module does'nt lift to an overconvergent filtration. As a by-product we show
that the unit-root sub-F-isocrystal of the de Rham cohomology of the Legendre
family of ordinary elliptic curves is not overconvergent in Berthelot's sense.
In the second article we'll give a definition of the L-functions of
F-(iso)crystals by cohomological means and we'll show how it matches with the
one given here: it gives back the one used in crystalline cohomology by Katz or
Etesse, or the one used in rigid cohomology by Etesse-Le Stum, or the one used
by Wan ; the aim is then to give a proof of Katz conjecture on p-adic unit
roots and poles of these L-functions using rigid cohomology. In the third
article we give an explicit form of these results for ordinary abelian schemes.
