$$ \frac{log(4750778730825177725463920948909726618214491718039471366318747406368792)}{ sqrt(652) } - \tau = -2.54282421310320265217436545223140117387 E-78 $$ found with Pari GP by playing with $e^{\tau \sqrt{163*4}}$ where $\pi := \frac{\tau}{2}$. See https://en.wikipedia.org/wiki/Almost_integer#Almost_integers_relating_to_e_and_%CF%80
Is this known?