What are some formulas involving Dottie number and digamma function?

Question

Here are two formulas that involve Dottie number and Gamma function:

$\Gamma \left( \frac{1}{2} + \frac{d}{\pi} \right) \Gamma \left( \frac{1}{2} - \frac{d}{\pi} \right) = \frac{\pi}{d} $

and

$\sqrt{\frac{1}{\pi^2}-\left(\frac{d}{\pi}\right)^2}\Gamma \left( \frac{1}{2} - \frac{d}{\pi} \right) \Gamma \left( \frac{d}{\pi} \right)=1$

Source.

Are there similar formulas involving digamma function?