| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 4·11-s + 14-s + 16-s + 4·17-s + 20-s + 4·22-s + 23-s + 25-s − 28-s + 6·29-s − 4·31-s − 32-s − 4·34-s − 35-s + 2·37-s − 40-s + 4·41-s + 4·43-s − 4·44-s − 46-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.685·34-s − 0.169·35-s + 0.328·37-s − 0.158·40-s + 0.624·41-s + 0.609·43-s − 0.603·44-s − 0.147·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.29450399229871, −16.02006946344678, −15.44790665832230, −14.70931141295773, −14.24819940131417, −13.54768421338562, −12.90782018683289, −12.52847046388239, −11.86712333813860, −11.05736410423775, −10.64036693319101, −9.994769366061367, −9.667523111966739, −8.930562735344017, −8.325744753032834, −7.634288162515254, −7.288749221893621, −6.301509741567078, −5.918946103920043, −5.150765837752496, −4.476854198248087, −3.253792241265305, −2.886683138762936, −1.980425412142563, −1.073064649884794, 0,
1.073064649884794, 1.980425412142563, 2.886683138762936, 3.253792241265305, 4.476854198248087, 5.150765837752496, 5.918946103920043, 6.301509741567078, 7.288749221893621, 7.634288162515254, 8.325744753032834, 8.930562735344017, 9.667523111966739, 9.994769366061367, 10.64036693319101, 11.05736410423775, 11.86712333813860, 12.52847046388239, 12.90782018683289, 13.54768421338562, 14.24819940131417, 14.70931141295773, 15.44790665832230, 16.02006946344678, 16.29450399229871
