Zero free region of Riemann zeta function

Question

For the zeros of the Riemann zeta function, $\rho=\beta+i \gamma$, is there a zero free region near $\beta=0$? I am familiar with the zero free regions near $\beta=1$, such as $\beta<1-1/\log(T)$ for $T>|\gamma|$, but I am mostly curious about the other end of the critical strip.