  For $0\leq \alpha<1$ and prime number $p$ let $L(\alpha,p)$ be the sum of the
first $[\alpha p]$ values of Legendre symbol modulo $p$. We study positivity of
$L(\alpha,p)$ and prove that for $|\alpha-\frac13|<2\cdot 10^{-6}$ and for
rational $\alpha\leq \frac12$ with denominators in the set
$\{1,2,3,4,5,6,8,12\}$ the inequality $L(\alpha,p)\geq 0$ holds for majority of
primes.
