The function $\text{Li}$ (logarithmic integral) is defined for $x>0$ by $$ \text{Li}(x)=\int_2^{x}\frac{dt}{\ln t}. $$ The prime number theorem, proven by Hadamard and de la Vallée-Poussin in 1896 asserts that $π(x) \sim \text{Li}(x)$ when $x$ goes to $+\infty$ where $ π(x)=\text{Card}\{2\le p\in \mathbb N, \text{ $p$ is prime}\}. $ It was proven by J. Littlewood in 1914 that the difference $d(x)=\text{Li}(x)-π(x)$ changes sign an infinite number of times, although for $x\le 10^{25}$, $d(x)>0$. (I guess that the $10^{25}$ could be enlarged.) I think that it is also known that for $x$ quite large, say in the interval $[10^{300}, 10^{500}]$, there exists $x$ such that $d(x)<0$. Thanks to the answers below, I have understood that there is no explicit value of $x$ for which we know that $d(x)<0$.

A related question: is it true that for $x$ large enough $$ π(x)\ge \frac{x}{\ln x}? $$