From the oscillating part of an explicit formula for primes:
$$\text{Re}(\operatorname{li}(x^\rho))\approx x\ \text{Re}(\operatorname{li}(x^{-\rho}))$$
$\rho_n$ may be replaced by any complex number, but the above expressions are closest in value only if real part $=\frac{1}{2}$. Is there any reason why this should be the case?
