Abstract
We incorporate the non-critical values of L-functions of cusp forms into a cohomological set-up analogous to the one of Eichler, Manin and Shimura. We use the 1-cocycles we associate in this way to non-critical values to prove an expression for such values which is similar in structure to Manin’s formula for the critical value of the L-function of a weight 2 cusp form.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Choie, Yj., Zagier, D.: Rational period functions for PSL(2, Z). Contemp. Math. 143, 89–108 (1993)
Diamantis, N.: Special values of higher derivatives of L-functions. Forum Math. 11, 229–252 (1999)
Diamantis, N.: Hecke operators and derivatives of L-functions. Compos. Math. 125(1), 39–54 (2001)
Diamantis, N., O’Sullivan, C.: Hecke theory of series formed with modular symbols and relations among convolution L-functions. Math. Ann. 318(1), 85–105 (2000)
Goldfeld, D.: Special values of derivatives of L-functions. Can. Math. Soc. Conf. Proc. 15, 159–173 (1995)
Knopp, M.: Rational period functions of the modular group. Duke Math. J. 45, 185–197 (1981)
Koblitz, N.: Non-integrality of the periods of cusp forms outside the critical strip. Funkcional. Anal. i Prilozhen. 9, 52–55 (1975)
Lawrence, R., Zagier, D.: Modular forms and quantum invariants of 3-manifolds. Asian J. Math. 3(1), 93–107 (1999)
Lewis, J.B., Zagier, D.: Period functions for modular forms of half-integral weight. I. Ann. Math. (2) 153(1), 191–258 (2001)
Manin, Y.T.: Parabolic points and zeta functions of modular forms. Izv. Akad. Nauk SSSR Ser. Mat. 6, 19–64 (1972)
Manin, Y.T.: Periods of cusp forms, and p-adic Hecke series. Mat. Sb. (N.S.) 92(134), 378–401 (1973)
Pribitkin, W.: Fractional integrals of modular forms. Acta Arith. 116(1), 43–62 (2005)
Zagier, D.: Hecke operators and periods of modular forms. Israel Math. Conf. Proc. 3, 321–336 (1990)
Zwegers, S.P.: Mock θ-functions and real analytic modular forms. In: q-Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000). Contemp. Math. 291 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
YoungJu Choie is partially supported by KOSEF R01-2003-00011596-0 and by ITRC Research Fund.
N. Diamantis is partially supported by EPSRC grant EP/D032350/1.
Rights and permissions
About this article
Cite this article
Choie, Y., Diamantis, N. Values of L-functions at integers outside the critical strip. Ramanujan J 14, 339–350 (2007). https://doi.org/10.1007/s11139-007-9034-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-007-9034-8