The ratio of the speed of the electron in the first Bohr orbit of hydrogen and the speed of light is equal to (where e, h and c have their usual meanings)

$2 \pi e^2 / h c$

Speed of electron in n^th orbit (in CGS) $v_n=\frac{2 \pi Z e^2}{n h}(k=1)$

For first orbit H₂ ; n = 1 and Z = 1

So $v=\frac{2 \pi e^2}{h} \Rightarrow \frac{v}{c}=\frac{2 \pi e^2}{h c}$