| L(s) = 1 | − 2-s + 2.61·3-s + 4-s − 2.61·6-s + 0.381·7-s − 8-s + 3.85·9-s + 2·11-s + 2.61·12-s + 2.61·13-s − 0.381·14-s + 16-s + 5.61·17-s − 3.85·18-s − 2·19-s + 21-s − 2·22-s − 1.85·23-s − 2.61·24-s − 2.61·26-s + 2.23·27-s + 0.381·28-s − 29-s + 2.85·31-s − 32-s + 5.23·33-s − 5.61·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.51·3-s + 0.5·4-s − 1.06·6-s + 0.144·7-s − 0.353·8-s + 1.28·9-s + 0.603·11-s + 0.755·12-s + 0.726·13-s − 0.102·14-s + 0.250·16-s + 1.36·17-s − 0.908·18-s − 0.458·19-s + 0.218·21-s − 0.426·22-s − 0.386·23-s − 0.534·24-s − 0.513·26-s + 0.430·27-s + 0.0721·28-s − 0.185·29-s + 0.512·31-s − 0.176·32-s + 0.911·33-s − 0.963·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.306189787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.306189787\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 7 | \( 1 - 0.381T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.61T + 13T^{2} \) |
| 17 | \( 1 - 5.61T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 1.85T + 23T^{2} \) |
| 31 | \( 1 - 2.85T + 31T^{2} \) |
| 37 | \( 1 + 8.94T + 37T^{2} \) |
| 41 | \( 1 + 5.70T + 41T^{2} \) |
| 43 | \( 1 - 5.61T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 3.85T + 59T^{2} \) |
| 61 | \( 1 + 4.14T + 61T^{2} \) |
| 67 | \( 1 - 8.94T + 67T^{2} \) |
| 71 | \( 1 - 7.70T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−9.334959039394387026234299613392, −8.667296058440451585561558864413, −8.153090784815264188905769993700, −7.42858905136872444202183355365, −6.55318092212780382572152232415, −5.48076939482947081225323196068, −3.99041399639357417512066942090, −3.36182028459217992485508310786, −2.27607005241651282330605319329, −1.26442527499546901267205251222,
1.26442527499546901267205251222, 2.27607005241651282330605319329, 3.36182028459217992485508310786, 3.99041399639357417512066942090, 5.48076939482947081225323196068, 6.55318092212780382572152232415, 7.42858905136872444202183355365, 8.153090784815264188905769993700, 8.667296058440451585561558864413, 9.334959039394387026234299613392
