Summary: The negative mode of the Schwarzschild black hole is central to Euclidean quantum gravity around hot flat space and for the Gregory-Laflamme black string instability. We analyze the eigenvalue as a function of spacetime dimension \(\lambda = \lambda (d)\) by constructing two perturbative expansions: one for large \(d\) and the other for small \(d - 3\), and determining as many coefficients as we are able to compute analytically. By joining the two expansions, we obtain an interpolating rational function accurate to better than 2% through the whole range of dimensions including \(d = 4\).