Who Let The Logarithms In?

Calculus Level 5

$\dfrac{\ln 4}{2^{3}} + \dfrac{\ln 6}{3^{3}} + \dfrac{\ln 8}{4^{3}} + \dfrac{\ln 10}{5^{3}} + \cdots$

You may use the approximations:
$\zeta(3)\approx 1.2020569$ and $\zeta'(3)\approx-0.198126$ .

Notation: $\zeta(\cdot)$ denotes the Riemann zeta function.