  Here we define the concept of $L$-regularity for coherent sheaves on the
Grassmannian G(1,4) as a generalization of Castelnuovo-Mumford regularity on
${\bf{P}^n}$. In this setting we prove analogs of some classical properties. We
use our notion of $L$-regularity in order to prove a splitting criterion for
rank 2 vector bundles with only a finite number of vanishing conditions. In the
second part we give the classification of rank 2 and rank 3 vector bundles
without "inner" cohomology (i.e. $H^i_*(E)=H^i(E\otimes\Q)=0$ for any
$i=2,3,4$) on G(1,4) by studying the associated monads.
