Is there an explicit formula connected to $(\log\zeta(s))^2$?

Is there an explicit formula connected to $(\log\zeta(s))^2$?

Riemann used $\log\zeta(s)$ and, essentially, Perron's formula to find the
explicit formula for his prime counting function, $\Pi(n)$:
$li(x)-\displaystyle\sum_{\rho}li(x^\rho)-\log
2-\int_{x}^{\infty}\frac{dt}{t(t^2-1)\log t}$
Is an explicit formula using $(\log\zeta(s))^2$ and Perron's formula to
compute the partial sum $\Pi_2(n) =
\displaystyle\sum_{j=2}^n\sum_{k=2}^{\lfloor \frac{n}{j}
\rfloor}\frac{\Lambda(j)}{ \log j}\frac{\Lambda(k)}{\log k}$ known?
(and here, $\Lambda(n)$ is the Von Mangoldt function)