| L(s) = 1 | + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s − 5·11-s − 13-s − 4·14-s + 16-s − 19-s + 20-s − 5·22-s + 23-s + 25-s − 26-s − 4·28-s + 29-s − 7·31-s + 32-s − 4·35-s − 7·37-s − 38-s + 40-s − 6·41-s + 11·43-s − 5·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s − 1.50·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.229·19-s + 0.223·20-s − 1.06·22-s + 0.208·23-s + 1/5·25-s − 0.196·26-s − 0.755·28-s + 0.185·29-s − 1.25·31-s + 0.176·32-s − 0.676·35-s − 1.15·37-s − 0.162·38-s + 0.158·40-s − 0.937·41-s + 1.67·43-s − 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−12.97822821605469, −12.70227215420848, −12.27526817421539, −11.97330549622973, −10.98850348158346, −10.75601326735716, −10.49564657082938, −9.733697186613355, −9.643895479611014, −8.944235514054340, −8.540177890911162, −7.765782429784578, −7.391325230661570, −6.987018002628080, −6.404991137157583, −5.999507522612854, −5.517111633274896, −5.164953552256511, −4.556794335069236, −3.966289017940060, −3.325610701850832, −2.967961055590817, −2.527092283249032, −1.952722997657572, −1.202429357372522, 0, 0,
1.202429357372522, 1.952722997657572, 2.527092283249032, 2.967961055590817, 3.325610701850832, 3.966289017940060, 4.556794335069236, 5.164953552256511, 5.517111633274896, 5.999507522612854, 6.404991137157583, 6.987018002628080, 7.391325230661570, 7.765782429784578, 8.540177890911162, 8.944235514054340, 9.643895479611014, 9.733697186613355, 10.49564657082938, 10.75601326735716, 10.98850348158346, 11.97330549622973, 12.27526817421539, 12.70227215420848, 12.97822821605469
