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We extend the method to find Hurwitz Zeta series of the form $S(k,a)=\\sum_{n \\in \\mathbb{Z}} \\frac{(-1)^{nk}}{(an+1)^k},$ where $a \\in \\mathbb{N} \\setminus \\lbrace 1 \\rbrace$ and $k \\in \\mathbb{N}.$ In particular, we consider a general $k$-dimensional integral over $(0,1)^k$ that equals the series representation $S(k,a).$ Then we use an algebraic change of variables that diffeomorphically maps $(0,1)^k$ to a $k$-dimensional hyperbolic polytope. 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