The general-\p formula for annealed Lyapunov exponents

Let $\xi$ be the random potential satisfying the essential-supremum and bounded independent-identically-distributed assumptions, let $\kappa>0$ be the diffusion parameter, and let $\lambda_p$ denote the $p$-th annealed Lyapunov exponent. For $h=1$ and $p\in(0,\infty)$, the general-\p Lyapunov-exponent conjecture. $\lambda_p$ is the zero of

$$\beta\mapsto\log\Big\langle\Big(\frac{\kappa}{\kappa+\beta-\xi(0)}\Big)^p\Big\rangle$$

in $(-\kappa,0)$ if this zero exists, and is $-\kappa$ otherwise. The preceding argument proves the formula only for natural $p$; extending it to all positive $p$ is expected but not established in the source.