6. Given complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}+z_{2}\right|=20,\left|z_{1}^{2}+z_{2}^{2}\right|=16$, then the minimum value of $\left|z_{1}^{3}+z_{2}^{3}\right|$ is
$$
\begin{array}{l}
\left|z_{1}^{3}+z_{2}^{3}\right|=\left|\left(z_{1}+z_{2}\right)\left(z_{1}^{2}+z_{2}^{2}-z_{1} z_{2}\right)\right|=10\left|3\left(z_{1}^{2}+z_{2}^{2}\right)-\left(z_{1}+z_{2}\right)^{2}\right| \\
\geqslant 10\left(\left|z_{1}+z_{2}\right|^{2}-3\left|z_{1}^{2}+z_{2}^{2}\right|\right)=10(400-3 \cdot 16)=3520,
\end{array}
$$

The equality holds when the vectors corresponding to $\left(z_{1}+z_{2}\right)^{2}$ and $z_{1}^{2}+z_{2}^{2}$ are in the same direction. For example, if $\left\{\begin{array}{l}z_{1}+z_{2}=20, \\ z_{1}^{2}+z_{2}^{2}=16\end{array} \Rightarrow z_{1} z_{2}=192\right.$, then $z_{1}, z_{2}$ are the roots of the equation $x^{2}-20 x+192=0$, solving which gives $z_{1}=10+2 \sqrt{23} \mathrm{i}, z_{2}=10-2 \sqrt{23} \mathrm{i}$.