We study Lorentzian hypersurfaces in the Lorentz-Minkowski 5-space E^5_1, which are defined by isometric immersions x : M^4_1 → E^5_1 satisfying the L_1-biharmonicity condition (L_1)^2 x = 0. The L_1-biharmonicity condition is an extension of the ordinary biharmonicity condition (i.e. (L_0)^2 x = 0) which has been studied by Bang-Yen Chen on the submanifolds of Euclidean spaces, where L0 = is the well-known Laplace operator. The operator L1 is the linearized map associated to the first variation of the second mean curvature vector field on M^4_1 . We discuss on Lorentzian hypersurfaces of E^5_1 having at most two distinct principal curvatures. After illustrating some examples, we prove that every L_1-biharmonic Lorentzian hypersurface with at most two distinct principal curvatures is 1-minimal.
