A conjectured value for $\operatorname{Re} \operatorname{Li}_4 (1 + i)$

by omegadot   Last Updated July 12, 2019 10:20 AM - source

In evaluating the integral given here it would seem that:

$$\operatorname{Re} \operatorname{Li}_4 (1 + i) \stackrel{?}{=} -\frac{5}{16} \operatorname{Li}_4 \left (\frac{1}{2} \right ) + \frac{97}{9216} \pi^4 + \frac{\pi^2}{48} \ln^2 2 - \frac{5}{384} \ln^4 2$$

I arrived at a result involving the $\operatorname{Re} \operatorname{Li}_4 (1 + i)$ term for the value of the integral while the OP is convinced the integral in question has a simple, elementary answer. If both of us are right, then the conjecture holds.

So my question is, is it possible to prove the conjecture is true or disprove the conjecture based on (very high precision) numerical evidence?

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