We exhibit a class of Littlewood polynomials that are not $L^α$-flat for any $α\geq 0$. Indeed, it is shown that the sequence of Littlewood polynomials is not $L^α$-flat, $α\geq 0$, when the frequency of $-1$ is not in the interval $]\frac14,\frac34[$. We further obtain a generalization of Jensen-Jensen-Hoholdt's result by establishing that the sequence of Littlewood polynomials is not $L^α$-flat for any $α> 2$ if the frequency of $-1$ is not $\frac12$. Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not $L^α$-flat for any $α\geq 0$.