A Lorentzian hypersurface x :M^4_1 ------>L^5 isometrically immersed into the Lorentz-Minkowski 5-space L^5, is said to be L_1-biconservative if the tangent component of vector field (L_1)^2 x is identically zero, where L1 is the linearized operator associated to the first variation of 2nd mean curvature vector field on M^4_1 . Since L_0 = D is the well known Laplace operator, The concept of L1-biconservative hypersurface is an extension of ordinary conservativity which is related to the physical concept of conservative stress-energy with respect to the bienergy functional. We discuss on Lorentzian hypersurfaces of L5 having at most two distinct principal curvatures. After illustrating some examples, we prove that every L_1-bicoservative Lorentzian hypersurface with constant ordinary mean curvature and at most two distinct principal curvatures in L^5 has to be of constant 2nd mean curvature.