JEE Main 2022 · 27 Jul · Shift 2, Mathematics question 28

Let $f$ be a differentiable function satisfying $f(x)=\dfrac2{\sqrt3}\displaystyle\int_0^{\sqrt3}f\left(\frac{\lambda^2x}3\right)d\lambda$, $x>0$ and $f(1)=\sqrt3$. If $y=f(x)$ passes through the point $(\alpha,6)$, then $\alpha$ is equal to ______.