We show that among all symmetric real log-concave random variables $X$ with variance 1 and any $t_0$ in range $[0,\sqrt3]$ the quantity $f_X(t_0)$ is minimized by a uniform, Laplace or truncated Laplace distribution. We show that for $t_0 \ge 1/\sqrt2$ the minimum is attained by Laplace distribution and for $t_0 \le 1/2$ it is attained by uniform distribution. We also show that the constant $1/\sqrt2$ cannot be improved and that there exist $t_0$ such that the minimizer is neither Laplace or uniform. This gives optimal dimension-free lower bounds for measures of non-central slices of isotropic convex bodies.