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Computing special values of motivic \(L\)-functions. (English) Zbl 1139.11317

Summary: We present an algorithm to compute values \(L(s)\) and derivatives \(L^{(k)}(s)\) of \(L\)-functions of motivic origin numerically to required accuracy. Specifically, the method applies to any \(L\)-series whose \(\Gamma\)-factor is of the form \(A^s\prod_{i=1}^d \Gamma(\frac{s+\lambda_j}{2})\) with \(d\) arbitrary and complex \(\lambda_j\), not necessarily distinct. The algorithm relies on the known (or conjectural) functional equation for \(L(s)\).

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11M41 Other Dirichlet series and zeta functions
11R42 Zeta functions and \(L\)-functions of number fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

Software:

ComputeL
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References:

[1] DOI: 10.1090/conm/055.1/862627 · doi:10.1090/conm/055.1/862627
[2] Birch B. J., J. Reine Angew. Math. 212 pp 7– (1963)
[3] Bloch S., The Grothendieck Festschrift pp 333– (1990)
[4] Braaksma B. L. J., Compositio Math. 15 pp 239– (1964)
[5] Buhler J. P., Math. Comp. 44 pp 473– (1985)
[6] DOI: 10.1007/978-1-4419-8489-0 · Zbl 0977.11056 · doi:10.1007/978-1-4419-8489-0
[7] DOI: 10.1098/rspa.1961.0187 · Zbl 0109.03102 · doi:10.1098/rspa.1961.0187
[8] Dokchitser T., ”ComputeL: Pari Package to Compute Motivic L-Functions.” (2002)
[9] Dokchitser T., ”Computations with L-functions of Higher Genus Curves.” · Zbl 1219.11082
[10] Fermigier Stéfane, Experimental Math. 1 (2) pp 167– (1992) · Zbl 0794.14006 · doi:10.1080/10586458.1992.10504254
[11] DOI: 10.1090/S0025-5718-1967-0240950-1 · doi:10.1090/S0025-5718-1967-0240950-1
[12] Henrici P., Applied and Computational Complex Analysis (1977) · Zbl 0363.30001
[13] Janssen U., Motives. (1994) · doi:10.1090/pspum/055.1
[14] DOI: 10.1090/S0025-5718-96-00734-X · Zbl 0853.11099 · doi:10.1090/S0025-5718-96-00734-X
[15] DOI: 10.1007/978-3-642-56478-9_10 · doi:10.1007/978-3-642-56478-9_10
[16] Lagarias J. C., Math. Comp. 33 (147) pp 1081– (1979)
[17] Lavrik A. F., Izv. Akad. Nauk SSSR 32 pp 134– (1968)
[18] Lorentzen L., Continued Fractions with Applications. (1992) · Zbl 0782.40001
[19] Luke Y. L., The Special Functions and Their Approximations 1 (1969) · Zbl 0193.01701
[20] Manin Yu. I., Encyclopedia of Math. Sciences pp 49– (1995)
[21] Rubinstein M. O., PhD diss., in: ”Evidence for a Spectral Interpretation of the Zeros of L-Functions.” (1998)
[22] DOI: 10.1017/CBO9780511526053.016 · doi:10.1017/CBO9780511526053.016
[23] Serre, J.P. ”Zeta andL-Functions.”. Arithmetical Algebraic Geometry, Proc. Conf. Purdue 1963. Edited by: Schilling, O. F. G. pp.82–92. New York: Harper & Row. [Serre 65]
[24] DOI: 10.2969/jmsj/02410132 · doi:10.2969/jmsj/02410132
[25] DOI: 10.1090/S0025-5718-97-00871-5 · Zbl 0877.11061 · doi:10.1090/S0025-5718-97-00871-5
[26] DOI: 10.1090/S0025-5718-1986-0829637-3 · doi:10.1090/S0025-5718-1986-0829637-3
[27] Yoshida H., J. Math. Kyoto Univ. 35 (4) pp 663– (1995) · Zbl 0865.11087 · doi:10.1215/kjm/1250518655
[28] DOI: 10.1007/978-1-4612-0457-2_19 · doi:10.1007/978-1-4612-0457-2_19
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