Let, $z_1$ and $z_2$ are roots of the equation $3z^2+3z+\lambda =0$
$\Rightarrow z_1+z_2=-1$ and $z_1{z}_2=\frac{\lambda }{3}$
If $0, z_1$ and $z_2$ form an equilateral triangle, then

$z_1^2+z_2^2=z_1{z}_2$
$\Rightarrow {\left(z_1+z_2\right)}^2=3z_1{z}_2$
$$\begin{gathered} \Rightarrow {\left(-1\right)}^2=3\left(\frac{\lambda }{3}\right)\Rightarrow \lambda =1 \\ \Rightarrow \left|z_1{z}_2\right|=\frac{1}{3} \end{gathered}$$

$\Rightarrow \sqrt{3}\left|z_1\right|=1$ unit