\(G\)はカタラン定数です。
$$
\begin {aligned}
\operatorname {Li} _2 \left (\sqrt 2-1\right )-\operatorname {Li} _2 \left (1-\sqrt 2\right )
&=\frac {\pi ^{2}}8-\frac {1}2\ln ^{2}\left (1+\sqrt 2\right )
\\
\operatorname {Li} _2 (i)
&=-\frac {\pi ^{2}}{48}+Gi
\\
\operatorname{Li} _{2}(1+i)
&=\frac {\pi ^{2}}{16}+\left (G+\frac {\pi \ln 2}4\right ) i
\\
\operatorname{Li} _{3}(i)
&=-\frac {3}{32}\zeta (3)+\frac {\pi ^{3}}{32}i
\\
\Re\operatorname{Li} _{3}\left (1+i\right )
&=\frac {35}{64}\zeta (3)+\frac {\pi ^{2}\ln 2}{32}
\\
\Re \operatorname{Li} _{3}\left (\frac {1+i}2\right )
&=\frac {35}{64}\zeta (3)-\frac {5\pi ^{2}\ln 2}{192}+\frac {\ln ^{3}2}{48}
\\
\operatorname{Li} _{3}\left (\frac {1+i}2\right ) +\operatorname{Li} _{3}(1+i)
&=\frac {35}{32}\zeta (3)+\frac {\pi ^{2}\ln 2}{192} +\frac {\ln ^{3}2}{48}\\
&+\left (\frac {7\pi ^{3}}{128}+\frac {3\pi \ln ^{2}2}{32}\right )i
\end {aligned}
$$