Let [.] denote the greatest integer function. If $\int_\limits0^{e^3}\left[\frac{1}{e^{x-1}}\right] d x=\alpha-\log _e 2$, then $\alpha^3$ is equal to _________.

If $ 24 \int\limits_0^{\frac{\pi}{4}} \bigg[\sin \left| 4x - \frac{\pi}{12} \right| + [2 \sin x] \bigg] dx = 2\pi + \alpha $, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.

Let $f:(0, \infty) \rightarrow \mathbf{R}$ be a twice differentiable function. If for some $a\ne 0, \int\limits_0^1 f(\lambda x) \mathrm{d} \mathrm{\lambda}=a f(x), f(1)=1$ and $f(16)=\frac{1}{8}$, then $16-f^{\prime}\left(\frac{1}{16}\right)$ is equal to __________.