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The polylogarithm $\rm{Li}_m\big(\tfrac12\big)$ has closed-forms known for $m=1,2,3$. It seems this triad pattern extends to the Nielsen generalized logarithm $S_{n,p}\,\big(\tfrac12\big)$ for $n=0,1,2$.

For brevity, let $\color{blue}{a= \ln 2}$. We then have,

For $p=1$

$$S_{0,1}(\tfrac12\big)=\rm{Li}_1\big(\tfrac12\big)=a \\ \qquad S_{1,1}(\tfrac12\big)=\rm{Li}_2\big(\tfrac12\big)=-\tfrac12a^2 +\tfrac12\zeta(2)\\ \qquad\quad S_{2,1}(\tfrac12\big)=\rm{Li}_3\big(\tfrac12\big)=\tfrac16a^3 -\tfrac12 a\,\zeta(2)+\tfrac78\zeta(3)\\$$

For $p=2$

$$S_{0,2}(\tfrac12\big)=\tfrac12 a^2 \\ S_{1,2}(\tfrac12\big)= -\tfrac16 a^3 +\tfrac18\zeta(3) \\ S_{2,2}(\tfrac12\big)= \tfrac1{24}a^4 -\tfrac18 a\,\zeta(3)+\tfrac18\zeta(4)$$

For $p=3$

$$S_{0,3}(\tfrac12\big)=\tfrac16 a^3 \\ S_{1,3}(\tfrac12\big)= -\tfrac16 a^4 -\tfrac12 a^2\rm{Li}_2\big(\tfrac12\big) - a\,\rm{Li}_3\big(\tfrac12\big) -\rm{Li}_4\big(\tfrac12\big)+\zeta(4)$$ $$S_{2,3}(\tfrac12\big)= \tfrac1{60}a^5 -\tfrac1{12} a^3\zeta(2)+\tfrac7{16}a^2\zeta(3)-\tfrac12\zeta(2)\zeta(3)-a\,\zeta(4)+\tfrac{63}{32}\zeta(5)-\rm{Li}_5\big(\tfrac12\big)$$

Q: Can we extend this "pattern" a bit more since for $p\ge 4$, the difficult evaluation is $S_{2,p}\,(\tfrac12\big)$?

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