Euclidean 5-space C^5, considered as the space of trace-free symmetric 3 x 3 complex matrices, has a natural twistor correspondence (it is in fact a typical example of a more general twistor correspondence, cf. Chapter 10 of). In this paper we work out a holomorphic Penrose transform, cf. (the familiarity with which will be assumed), associated with it. For techniques used in analyzing data on a holomorphic homogeneous vector bundle, see e.g. for some preliminaries.