6. Given complex numbers $z_{1}, z_{2}$ satisfy
$$
\left|z_{1}+z_{2}\right|=20,\left|z_{1}^{2}+z_{2}^{2}\right|=16 \text {. }
$$

Then the minimum value of $\left|z_{1}^{3}+z_{2}^{3}\right|$ is $\qquad$
6. 3520 .

Notice that,
$$
\begin{array}{l}
\left|z_{1}^{3}+z_{2}^{3}\right|=\left|z_{1}+z_{2}\right|\left|z_{1}^{2}-z_{1} z_{2}+z_{2}^{2}\right| \\
=20\left|\frac{3}{2}\left(z_{1}^{2}+z_{2}^{2}\right)-\frac{1}{2}\left(z_{1}+z_{2}\right)^{2}\right| \\
\left.\geqslant 20\left|\frac{3}{2}\right| z_{1}^{2}+z_{2}^{2}\left|-\frac{1}{2}\right| z_{1}+\left.z_{2}\right|^{2} \right\rvert\, \\
=3520 .
\end{array}
$$

When $z_{1}^{2}+z_{2}^{2}=16, z_{1}+z_{2}=20$, i.e.,
$$
z_{1}=10+2 \sqrt{23} \mathrm{i}, z_{2}=10-2 \sqrt{23} \mathrm{i}
$$

the equality in (1) holds.
Therefore, the minimum value of $\left|z_{1}^{3}+z_{2}^{3}\right|$ is 3520.