| L(s) = 1 | − 4·2-s + 23.4·3-s + 16·4-s − 93.9·6-s − 49·7-s − 64·8-s + 308.·9-s − 334.·11-s + 375.·12-s − 420.·13-s + 196·14-s + 256·16-s + 1.12e3·17-s − 1.23e3·18-s − 352.·19-s − 1.15e3·21-s + 1.33e3·22-s − 737.·23-s − 1.50e3·24-s + 1.68e3·26-s + 1.53e3·27-s − 784·28-s − 4.35e3·29-s − 1.41e3·31-s − 1.02e3·32-s − 7.85e3·33-s − 4.50e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.50·3-s + 0.5·4-s − 1.06·6-s − 0.377·7-s − 0.353·8-s + 1.26·9-s − 0.834·11-s + 0.753·12-s − 0.690·13-s + 0.267·14-s + 0.250·16-s + 0.945·17-s − 0.897·18-s − 0.224·19-s − 0.569·21-s + 0.589·22-s − 0.290·23-s − 0.532·24-s + 0.488·26-s + 0.405·27-s − 0.188·28-s − 0.961·29-s − 0.264·31-s − 0.176·32-s − 1.25·33-s − 0.668·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 - 23.4T + 243T^{2} \) |
| 11 | \( 1 + 334.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 420.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.12e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 352.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 737.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.35e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.41e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.63e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.12e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.53e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.79e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.10e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.04e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.61e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.07e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.08e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.57e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.33e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.49e5T + 8.58e9T^{2} \) |
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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
−9.797242282717251427288529185299, −9.381195232268584584230833713936, −8.172766390383903786873173212880, −7.81495362277107638039334416980, −6.73637022003719321348971907216, −5.25416244777248025451176385760, −3.62916266070714246110292941825, −2.75966703695891764166588559157, −1.75102095239761784055911271036, 0,
1.75102095239761784055911271036, 2.75966703695891764166588559157, 3.62916266070714246110292941825, 5.25416244777248025451176385760, 6.73637022003719321348971907216, 7.81495362277107638039334416980, 8.172766390383903786873173212880, 9.381195232268584584230833713936, 9.797242282717251427288529185299
