  A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all of its
coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due
to Littlewood and Erd\H{o}s, that ask for Littlewood polynomials that provide a
good approximation to a function that is constant on the complex unit circle,
and in particular have small $L^q$ norm on the complex unit circle. We consider
the Fekete polynomials \[ f_p(z)=\sum_{j=1}^{p-1}(j\mid p)\,z^j, \] where $p$
is an odd prime and $(\,\cdot\mid p)$ is the Legendre symbol (so that
$z^{-1}f_p(z)$ is a Littlewood polynomial). We give explicit and recursive
formulas for the limit of the ratio of $L^q$ and $L^2$ norm of $f_p(z)$ when
$q$ is an even positive integer and $p\to\infty$. To our knowledge, these are
the first results that give these limiting values for specific sequences of
nontrivial Littlewood polynomials and infinitely many $q$. Similar results are
given for polynomials obtained by cyclically permuting the coefficients of
Fekete polynomials and for Littlewood polynomials whose coefficients are
obtained from additive characters of finite fields. These results vastly
generalise earlier results on the $L^4$ norm of these polynomials.
