Archimedean local $L$-factors were introduced to simplify functional equations of global $L$-functions. From the point of view of arithmetic geometry these factors complete the Euler product representation of global $L$-factors by taking into account Archimedean places of the compactified spectrum of the global field. A construction of non-Archimedean local $L$-factors is rather transparent and uses characteristic polynomial of the image of the Frobenius homomorphism in finite-dimensional representations of the local Weil-Deligne group closely related to the local Galois group. On the other hand, Archimedean $L$-factors are expressed through products of $Gamma$-functions and thus are analytic objects avoiding simple algebraic interpretation. In a series of papers [1,2,3,4] we approach the problem of the proper interpretation of Archimedean $L$-factors using various methods developed to study quantum integrable systems and low-dimensional topological field theories. As a result we produce several interesting explicit representations for Archimedean $L$-factors and related special functions revealing some hidden structures that might be relevant to the Archimedean (also known as $infty$-adic ) algebraic geometry. par noindent The talk is based on common with A.Gerasimov and S. Kharchev papers: par noindent [1] A.~Gerasimov, D.~Lebedev, S.~Oblezin, {it Baxter operator and Archimedean Hecke algebra}, Comm. Math. Phys. DOI 10.1007/s00220-008-0547-9; {tt [arXiv:0706.347]}, 2007. par noindent [2] A.~Gerasimov, D.~Lebedev, S.~Oblezin, {it Baxter Q-operators and their Arithmetic implications}, Lett. Math. Phys. DOI 10.1007/911005-008-0285-0; {tt [arXiv:0711.2812]}. par noindent [3] A.~Gerasimov, D.~Lebedev, S.~Oblezin, {it On q-deformed $mathfrak{gl}_{ell+1}$-Whittaker functions I,II,III}, {tt [arXiv:0803.0145]}, {tt [arXiv:0803.0970]}, {tt [arXiv:0805.3754]}. par noindent [4] A.~Gerasimov, D.~Lebedev, S.~Oblezin, {it Archimedean L-factors and Topological Field Theories}, {tt [arXiv:0906.1065]}.

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