This paper considers absolute stability for Lur'e systems consisting of the interconnection of a linear plant with a nonlinear feedback. The nonlin-earity is assumed to be both sector bounded and slope restricted. Stability of this system is determined using a Lyapunov function with a quadratic term on both the states and the nonlinearity. The main result of this paper is to relax the positivity conditions that have been imposed for such Lyapunov functions. This allows Lyapunov functions to be constructed without a positive definite quadratic matrix and whose scalars of the Lur'e term are not sign-definite. We also show that previous results can be simplified to the case of the quadratic form with Lur'e term. The benefits of considering such a Lyapunov function for stability analysis are shown both for the global case and for the local case.