We know that Skewes number is the smallest integer $x$ such that,
$$\pi(x) > \operatorname{li}(x)$$
where $\pi(x)$ is the prime counting function and $\operatorname{li}(x)$ is the logarithmic integral function.
The same Wikipedia article explains that the current best estimate is around $1.39716 \times 10^{316}$ and that $\pi(x) - \operatorname{li}(x)$ switches signs infinitely often.
It is sometimes suggested that $\operatorname{li}(x) - \frac{1}{2}\operatorname{li}(\sqrt{x})$ is a better estimate for $\pi(x)$. But given the above, this can’t always be true although it is certainly true for small $x$.
What is a good estimate for the smallest $x$ for which
$$|\operatorname{li}(x) - \frac{1}{2}\operatorname{li}(\sqrt{x}) - \pi(x)| > | \operatorname{li}(x) - \pi(x) |\;?$$
By the research following Littlewood and Skewes, it's known to be less than $e^{727.95133}$. But we can do better with some direct calculation!
Excluding $x\in[2,2.106472\ldots)$, the first example that I find is $x=30902129$. If I'm not mistaken this is the first crossover with $x\ge3.$