This paper deals with the stability analysis of the reaction-diffusion equation interconnected with a finite-dimensional system. In this situation, stability is no longer straightforward to assess, and one needs to look for dedicated tools to provide accurate numerical stability tests. Here, the objective is to provide a Lyapunov analysis leading to efficient and precise stability criteria. Thanks to the Legendre orthogonal basis, the study is possible using dedicated Lyapunov functionals. Such functionals contain information based on the state of an augmented finite-dimensional system and a remaining infinite-dimensional system, associated with the first Legendre coefficients and the remainder of the Legendre expansion, respectively. Furthermore, the Legendre's development provides efficient formulations of Bessel and Wirtinger inequalities and leads to sufficient stability conditions expressed as scalable linear matrix inequalities. Numerical examples finally illustrate the accuracy and the potential of such stability results.