(eng): Space-time regularity of linear stochastic partial differential equations is studied. The solution is defined in the mild sense in the state space Lp. The corresponding regularity is obtained by showing that the stochastic convolution integrals are Hölder continuous in a suitable function space. In particular cases, this allows us to show space-time Hölder continuity of the solution. The main tool used is a hypercontractivity result on Banach-space valued random variables in a finite Wiener chaos.