Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a twice differentiable function. If for some $a \neq 0, \int_0^1 f(\lambda x) d\lambda = a f(x)$, $f(1) = 1$ and $f(16) = \frac{1}{8}$, then $16 - f' \left( \frac{1}{16} \right)$ is equal to ______.