Limit of Series with differences of Floor function
With the change of variable $x=\frac{1}{z}$ we get:
$$ L = \int_{1}^{+\infty}\frac{\lfloor 2z\rfloor -2\lfloor z\rfloor}{z^2}\,dx =\sum_{k\geq 1}\int_{k+\frac{1}{2}}^{k+1}\frac{dz}{z^2}=\sum_{k\geq 1}\frac{2}{(2k+1)(2k+2)}$$ hence:
$$ L = 2\int_{0}^{1}\sum_{k\geq 1}\left(z^{2k}-z^{2k+1}\right)\,dz =\int_{0}^{1}\frac{2z^2}{1+z}\,dz=\color{red}{2\log 2-1}.$$