<<PolylogTools`

In1[n_Integer /; n ≥ 1, x_,y_]:=G@@Join[{y} , Table[0, {n-1}], {x, 1}]

Li[{5}, {x_}] := -G[0, 0, 0, 0, 1/x, 1]
Li[{6}, {x_}] := -G[0, 0, 0, 0, 0, 1/x, 1]
LiM[n_, x_, y_] := Sum[(-Log[x])^i Li[{n-i},{y}]/i!,{i,0,n-1}]

identity[n_Integer /; n ≥1] := In1[n, x, y] + (-1)^n In1[n, 1/x, 1/y]+
(-1)^n (Li[{n+1},{x}]+n Li[{n+1},{x/y}])+
LiM[n+1, x, y] - 
(-1)^n Log[x/y] Li[{n}, {x/y}] -
(-1)^n(Log[x/y]^(n+1) - Log[x]^(n+1))/(n+1)!

(* --------- *)
tolog[expr_] := ((expr)/.G[a_,b_]:>GLog[a,b])

expandlog[expr_] := ((expr)/.Log[a_]:>Log[Factor[a]]/.Log[a_Times]:> (Plus@@(Log/@(List@@a)))/.Log[a_^b_]:>b Log[a])

canonicallog[expr_] := ((expr)/.{Log[-1+x]:>(I Pi + Log[1-x]),Log[-1+y]:>(I Pi + Log[1-y]),Log[-x+y]:>(I Pi + Log[x-y])})

(* --------- *)

id = identity[1]

delta1 = Delta11[id]//tolog//expandlog//canonicallog

(delta1/(-I)//Expand)/.{CT[Pi, rest__]:>CT[rest]}//ToSymbol

delta1corr = I Pi (Log[x] + Li[{1}, {y}])

id1 = identity[1] + delta1corr

Delta11[id1]//tolog//expandlog//canonicallog

N[id1/.{Li[{n_}, {x_}]:>PolyLog[n, x],G->GLog}/.{x->-165+12I,y->-154-17I}, 20]

N[2Pi^2/3, 20]

idfull = identity[1] + delta1corr + 2Pi^2/3

N[idfull/.{Li[{n_}, {x_}]:>PolyLog[n, x],G->GLog}/.{x->-165+12I,y->-154-17I}, 20]