| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s + 4·11-s − 12-s − 6·13-s − 15-s + 16-s − 2·17-s + 18-s + 4·19-s + 20-s + 4·22-s + 8·23-s − 24-s + 25-s − 6·26-s − 27-s − 6·29-s − 30-s + 32-s − 4·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s − 1.66·13-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.852·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 1.11·29-s − 0.182·30-s + 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.575439374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.575439374\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.95435804425530, −14.61454892434203, −14.30856042325556, −13.48118322851211, −13.04884866847220, −12.55533720277488, −11.96323153405391, −11.59221718946110, −11.00984172958980, −10.50921392231750, −9.687514305366521, −9.323943140610091, −8.910577170740961, −7.634485827608637, −7.377262708025919, −6.787697210718871, −6.186642997025166, −5.602616376298481, −4.964318907380310, −4.572779176850053, −3.821856684837905, −3.009540621240462, −2.354655860771129, −1.522623510655243, −0.6922596533629135,
0.6922596533629135, 1.522623510655243, 2.354655860771129, 3.009540621240462, 3.821856684837905, 4.572779176850053, 4.964318907380310, 5.602616376298481, 6.186642997025166, 6.787697210718871, 7.377262708025919, 7.634485827608637, 8.910577170740961, 9.323943140610091, 9.687514305366521, 10.50921392231750, 11.00984172958980, 11.59221718946110, 11.96323153405391, 12.55533720277488, 13.04884866847220, 13.48118322851211, 14.30856042325556, 14.61454892434203, 14.95435804425530
