This paper is the third of a series on semigroups of operator related to the Laplace Beltrami operator on a symmetric space of the non compact type. We consider the Poisson semigroup P_{τ,θ}, when θ=1 and τ is complex and Reτ>0. We remark that the shifted Laplace Beltrami operator -L+b, corresponding to the case θ=1, occurs naturally in geometry, as it is conformally invariant. Our main theorem describes the behaviour of the Lp-Lq operator norm of P_{τ,1} for various possible values of p and q and for τ in various subsets of the right half of the complex plane. This description is nearly complete, but when p<2<q and |τ| is large but τ is nearly imaginary, our methods do not yield good estimates.