Exact bounds on the distribution function of isotropic log-concave distributions

For each real $b$, exact upper and lower bounds on the probability $\mathsf{P}(X\ge b)$ over all random variables $X$ with log-concave p.d.f.'s such that $\mathsf{E}X=0$ and $\mathsf{E}{X}^2=1$ are obtained, as well as the best constant factor $C$ in the inequality $\mathsf{P}(X\ge b)\le C{e}^{-b}$ for all real $b\ge 0$. Explicit exponentially decreasing upper bounds on the mentioned p.d.f.'s are given as well. Some general results concerning log-concave p.d.f.'s are also obtained.