normalized Laplacian of Gaussian Laplacian of Gaussian formula for 2d case is
$$\operatorname{LoG}(x,y) = \frac{1}{\pi\sigma^4}\left(\frac{x^2+y^2}{2\sigma^2} - 1\right)e^{-\frac{x^2+y^2}{2\sigma^2}},$$ in scale-space related processing of digital images, to make the Laplacian of Gaussian operator invariant to scales, it is always said to normalize $LoG$ by multiplying $\sigma^2$, that is
$$\operatorname{LoG}_\text{normalized}(x,y) = \sigma^2\cdot \operatorname{LoG}(x,y) = \frac{1}{\pi\sigma^2}\left(\frac{x^2+y^2}{2\sigma^2} - 1\right)e^{-\frac{x^2+y^2}{2\sigma^2}}.$$ I wonder why multiply by $\sigma^2$ not $\sigma^4$ or anything else?
Laplacian response decays as $\sigma$ increases, and it is the second Gaussian derivative so it is multiplied by $\sigma^2$.

LOG is defined as $\bigtriangledown^2G$ so the scale normalized LOG would be $\sigma^2\bigtriangledown^2G$. You need to get rid of the scaling factor of the Gaussian which is $\sigma^2$.