  In this paper, we study \lambda-biharmonic hypersurfaces in the product space
L^{m}\times\mathbb{R}, where L^{m} is an Einstein space and \mathbb{R} is a
real line. We prove that \lambda-biharmonic hypersurfaces with constant mean
curvature in L^{m}\times\mathbb{R} are either minimal or vertical cylinders,
and obtain some classification results for \lambda$-biharmonic hypersurfaces
under various constraints. Furthermore, we investigate \lambda-biharmonic
hypersurfaces in the product space L^{m}(c)\times\mathbb{R}, where L^{m}(c) is
a space form with constant sectional curvature c, and categorize hypersurfaces
that are either totally umbilical or semi-parallel.
