Damned Numbers

Carl and kneemo, amongst others, like to think about that damned number, otherwise known as the phase angle determining the charged lepton mass matrix, which is

$\phi =0.22222204717$

to within experimental precision: notably close to $\frac{2}{9}$. The $3\times 3$ MUB problem says nothing about this phase. Since phases usually involve factors of $\pi$, one wonders if there are any well known numbers that, when multiplied by $\pi$, also give numbers very close to $\frac{2}{9}$. For example, consider the first zero of the Riemann zeta function, namely ${\gamma }_1=14.134725142$. Observe that

$\frac{\pi }{{\gamma }_1}=0.222260611\left(5\right)$

which differs from $\frac{2}{9}$ by a factor of 1.000172751(75). So we didn't really need to look far to find a number satisfying this curiosity. Are there better ones?