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We consider the space X = Sym^3(C^2) of binary cubic forms, equipped with the natural action of the group GL_2 of invertible linear transformations of C^2. We describe explicitly the category of GL_2-equivariant coherent D_X-modules as the category of representations of a quiver with relations. We show moreover that this quiver is of tame representation type and we classify its indecomposable representations. We also give a construction of the simple equivariant D_X-modules (of which there are 14), and give formulas for the characters of their underlying GL_2-representations. We conclude the article with an explicit calculation of (iterated) local cohomology groups with supports given by orbit closures.