$\alpha =\frac{F{v}^2}{{\beta }^2}{\log }_e\left(\frac{2\pi \beta }{v^2}\right)$ where F = force, v = velocity
Find the dimensions of $\alpha$ and $\beta$

Logarithm functions are dimensionless

$\therefore \left[\frac{2\pi \beta }{V^2}\right]=M^0{L}^0{T}^0\left[\beta \right]=\left[V^2\right]={\left(L{T}^{-1}\right)}^2=L^2{T}^{-2}$

From relation,

$\left[\alpha \right]=\left[\frac{F{V}^2}{{\beta }^2}\right]=\frac{ML{T}^{-2}.{\left(L{T}^{-1}\right)}^2}{{\left(L^2{T}^{-2}\right)}^2}=\frac{ML{T}^{-2}.L^2{T}^{-2}}{L^4{T}^{-4}}=M{L}^{-1}{T}^0$