| L(s) = 1 | + 9·3-s + 4·5-s + 49·7-s + 81·9-s − 370·11-s + 122·13-s + 36·15-s − 1.42e3·17-s − 1.72e3·19-s + 441·21-s + 2.67e3·23-s − 3.10e3·25-s + 729·27-s + 4.30e3·29-s − 3.10e3·31-s − 3.33e3·33-s + 196·35-s − 1.43e4·37-s + 1.09e3·39-s − 1.22e4·41-s + 2.16e4·43-s + 324·45-s + 2.64e3·47-s + 2.40e3·49-s − 1.28e4·51-s − 2.43e4·53-s − 1.48e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.0715·5-s + 0.377·7-s + 1/3·9-s − 0.921·11-s + 0.200·13-s + 0.0413·15-s − 1.19·17-s − 1.09·19-s + 0.218·21-s + 1.05·23-s − 0.994·25-s + 0.192·27-s + 0.949·29-s − 0.580·31-s − 0.532·33-s + 0.0270·35-s − 1.71·37-s + 0.115·39-s − 1.14·41-s + 1.78·43-s + 0.0238·45-s + 0.174·47-s + 1/7·49-s − 0.691·51-s − 1.19·53-s − 0.0659·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 4 T + p^{5} T^{2} \) |
| 11 | \( 1 + 370 T + p^{5} T^{2} \) |
| 13 | \( 1 - 122 T + p^{5} T^{2} \) |
| 17 | \( 1 + 84 p T + p^{5} T^{2} \) |
| 19 | \( 1 + 1724 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2670 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4302 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3104 T + p^{5} T^{2} \) |
| 37 | \( 1 + 14318 T + p^{5} T^{2} \) |
| 41 | \( 1 + 12272 T + p^{5} T^{2} \) |
| 43 | \( 1 - 21652 T + p^{5} T^{2} \) |
| 47 | \( 1 - 2644 T + p^{5} T^{2} \) |
| 53 | \( 1 + 24342 T + p^{5} T^{2} \) |
| 59 | \( 1 - 14088 T + p^{5} T^{2} \) |
| 61 | \( 1 + 24474 T + p^{5} T^{2} \) |
| 67 | \( 1 + 7208 T + p^{5} T^{2} \) |
| 71 | \( 1 + 54302 T + p^{5} T^{2} \) |
| 73 | \( 1 + 48962 T + p^{5} T^{2} \) |
| 79 | \( 1 - 33332 T + p^{5} T^{2} \) |
| 83 | \( 1 - 4004 T + p^{5} T^{2} \) |
| 89 | \( 1 - 64752 T + p^{5} T^{2} \) |
| 97 | \( 1 - 7038 T + p^{5} T^{2} \) |
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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
−10.41106495831459652142528218698, −9.114860015458025569745290560572, −8.472956935696636659282138064945, −7.49821148056134348240584405255, −6.47138630085650071421124627890, −5.16099003866897847545593450427, −4.14829009853428805835301431943, −2.78864018904290743236501555468, −1.75013417284715999866801489945, 0,
1.75013417284715999866801489945, 2.78864018904290743236501555468, 4.14829009853428805835301431943, 5.16099003866897847545593450427, 6.47138630085650071421124627890, 7.49821148056134348240584405255, 8.472956935696636659282138064945, 9.114860015458025569745290560572, 10.41106495831459652142528218698
