Let $\lambda \in R,$  the origin and the non-real roots of $2z^2+2z+\lambda =0$  from the three vertices of an equilateral triangle in the Argand plane then $\lambda$  is

$2z^2+2z+\lambda =0$ 

Let the roots be $z_1,z_2.$  Then, $z_1+z_2=-1$  and $z_1{z}_2=\frac{\lambda }{2}$

$0,z_1,z_2$  form an equilateral triangle.   $\left(if z_{1,}{z}_{2,}{z}_{3 } form an equilateral triangle then {z_1}^2+{z_2}^2+{z_3}^2=z_1{z}_2+z_2{z}_3+z_3{z}_1\right)$

$\therefore$  $z_1^2+z_2^2=z_1{z}_2$

$\Rightarrow$  ${(z_1+z_2)}^2=3z_1{z}_2$

$\Rightarrow$  $1=3\frac{\lambda }{2}$

$\Rightarrow$  $\lambda =\frac{2}{3}$