L(s) = 1 | + (0.791 − 0.497i)2-s + (0.365 − 0.930i)3-s + (−0.0549 + 0.114i)4-s + (−0.173 − 0.918i)6-s + (0.117 + 1.04i)8-s + (−0.733 − 0.680i)9-s + (0.0861 + 0.0928i)12-s + (1.49 − 1.19i)13-s + (0.534 + 0.670i)16-s + (−0.918 − 0.173i)18-s + (0.974 − 0.222i)23-s + (1.01 + 0.272i)24-s + (−0.900 − 0.433i)25-s + (0.589 − 1.68i)26-s + (−0.900 + 0.433i)27-s + ⋯ |
L(s) = 1 | + (0.791 − 0.497i)2-s + (0.365 − 0.930i)3-s + (−0.0549 + 0.114i)4-s + (−0.173 − 0.918i)6-s + (0.117 + 1.04i)8-s + (−0.733 − 0.680i)9-s + (0.0861 + 0.0928i)12-s + (1.49 − 1.19i)13-s + (0.534 + 0.670i)16-s + (−0.918 − 0.173i)18-s + (0.974 − 0.222i)23-s + (1.01 + 0.272i)24-s + (−0.900 − 0.433i)25-s + (0.589 − 1.68i)26-s + (−0.900 + 0.433i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.923692505\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.923692505\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.365 + 0.930i)T \) |
| 23 | \( 1 + (-0.974 + 0.222i)T \) |
| 29 | \( 1 + (-0.997 + 0.0747i)T \) |
good | 2 | \( 1 + (-0.791 + 0.497i)T + (0.433 - 0.900i)T^{2} \) |
| 5 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 7 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 13 | \( 1 + (-1.49 + 1.19i)T + (0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 31 | \( 1 + (0.275 + 0.438i)T + (-0.433 + 0.900i)T^{2} \) |
| 37 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 41 | \( 1 + (0.839 - 0.839i)T - iT^{2} \) |
| 43 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 47 | \( 1 + (1.50 + 0.169i)T + (0.974 + 0.222i)T^{2} \) |
| 53 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 59 | \( 1 - 1.80iT - T^{2} \) |
| 61 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 67 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 71 | \( 1 + (-0.848 - 1.06i)T + (-0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (0.197 - 0.314i)T + (-0.433 - 0.900i)T^{2} \) |
| 79 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 83 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 97 | \( 1 + (0.781 - 0.623i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.859144946730870379005898111556, −8.321675970660408747651684572838, −7.81975378339190298728951440070, −6.72905590791436486609770086105, −5.93203328759352493371688061281, −5.17876601990079263269488525696, −3.97703321149938455316914157493, −3.24607763075991372930367846297, −2.50666355198014357605595258002, −1.24253000687596509081482811039,
1.63664517257999090804666446787, 3.26171499667276643962191552165, 3.86180572741484170946016351045, 4.67218080684964490290722804189, 5.35715657712781248340577153596, 6.24070498254304326273146431567, 6.87423214057207944592621816506, 8.022700914601131312913598940598, 8.899532518566600353445846355797, 9.381890021501511579361940647775