For V a 2-dimensional p-adic representation of G_Qp, we denote by B(V) the
admissible unitary representation of GL_2(Qp) attached to V under the p-adic
local Langlands correspondence of GL_2(Qp) initiated by Breuil. In this
article, building on the works of Berger-Breuil and Colmez, we determine the
locally analytic vectors B(V)an of B(V) when V is irreducible, crystabelian and
Frobenius semi-simple with Hodge-Tate weights (0,k-1) for some integer k>=2;
this proves a conjecture of Breuil. Using this result, we verify Emerton's
conjecture that dim Ref^{\eta\otimes\psi}(V)=dim Exp^{\eta|\cdot|\otimes
x\psi}(B(V)an\otimes(x|\cdot|\circ\det)) for those V which are irreducible,
crystabelian and not exceptional.