We prove that for a certain class of n dimensional rank one locally symmetric spaces, if $$f \in L^p$$ , $$1\le p \le 2$$ then the Riesz means order z f converge to almost everywhere, $$\mathrm {Re}z> (n-1)(1/p-1/2)$$ .

We study the L-spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M = Γ\X with finite volume and arithmetic fundamental group Γ whose universal covering X is a symmetric space of non-compact type. We also show, how the obtained results for locally symmetric spaces can be generalized to manifolds with cusps of rank one.