EN
We study the family of curves $F_{m}(p): x^{p} + y^{p} = m$, where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves $F_{m}(p)$. As a corollary we conclude that the jacobians of the curves $F_{m}(5)$ with even analytic rank and those with odd analytic rank are equally distributed.