Geodesic
Orbital $L$ -functions for the Space of Binary Cubic Forms
Canadian journal of mathematics, Tome 65 (2013) no. 6, pp. 1320-1383

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DOI

We introduce the notion of orbital $L$ -functions for the space of binary cubic forms and investigate their analytic properties. We study their functional equations and residue formulas in some detail. Aside from their intrinsic interest, the results from this paper are used to prove the existence of secondary terms in counting functions for cubic fields. This is worked out in a companion paper.
DOI : 10.4153/CJM-2013-027-0
Mots-clés : 11M41, 11E76, binary cubic forms, prehomogeneous vector spaces, Shintani zeta functions, L-functions, cubic rings and fields 
Taniguchi, Takashi; Thorne, Frank. Orbital $L$ -functions for the Space of Binary Cubic Forms. Canadian journal of mathematics, Tome 65 (2013) no. 6, pp. 1320-1383. doi: 10.4153/CJM-2013-027-0
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     title = {Orbital $L$ -functions for the {Space} of {Binary} {Cubic} {Forms}},
     journal = {Canadian journal of mathematics},
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     year = {2013},
     volume = {65},
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     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2013-027-0/}
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