  Let C be a smooth projective curve defined over a number field and let C' be
a twist of C. In this article we relate the l-adic representations attached to
the l-adic Tate modules of the Jacobians of C and C' through an Artin
representation. This representation induces global relations between the local
factors of the respective Hasse-Weil L-functions. We make these relations
explicit in a particularly illustrating situation. For every Qbar-isomorphism
class of genus 2 curves defined over Q with automorphism group isomorphic to
D_8 or D_{12}, except for a finite number, we choose a representative curve C/Q
such that, for every twist C' of C satisfying some mild condition, we are able
to determine either the local factor L_p(C'/Q,T) or the product
L_p(C'/Q,T)L_p(C'/Q,-T) from the local factor L_p(C/Q,T).
