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Calculus Level 5

Lis1,s2,,sk(z1,z2,,zk)=n1>n2>>nk>0j=1knjsjzjnj\displaystyle \text{Li}_{s_1,s_2,\dots,s_k}(z_1,z_2,\dots,z_k)=\sum_{n_1>n_2>\dots>n_k>0}\prod_{j=1}^{k}n_j^{-s_j}z_j^{n_j}

Multiple polylogarihtms are multiply nested sums of the form above. Given that

Li1,2(12,1)=lnABζ(C)D(lnF)EG,\displaystyle \text{Li}_{1,2} \left(\dfrac12,1\right)=\dfrac{\ln A}{B} - \dfrac{\zeta(C)}{D} - \dfrac{(\ln F)^E}{G} ,

where A,B,C,D,E,FA,B,C,D,E,F and GG are integers, find the minimum value of A+B+C+D+E+F+GA+B+C+D+E+F+G.

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