  Space-time regularity of linear stochastic partial differential equations is
studied. The solution is defined in the mild sense in the state space $L^p$.
The corresponding regularity is obtained by showing that the stochastic
convolution integrals are H\"{o}lder continuous in a suitable function space.
In particular cases, this allows to show space-time H\"{o}lder continuity of
the solution. The main tool used is a hypercontractivity result on Banach-space
valued random variables in a finite Wiener chaos.
