| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s + 13-s + 15-s − 17-s + 4·19-s − 2·21-s + 25-s + 27-s + 6·29-s − 10·31-s − 2·35-s − 4·37-s + 39-s − 2·41-s + 4·43-s + 45-s − 6·47-s − 3·49-s − 51-s − 6·53-s + 4·57-s − 6·59-s + 6·61-s − 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.917·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.79·31-s − 0.338·35-s − 0.657·37-s + 0.160·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.140·51-s − 0.824·53-s + 0.529·57-s − 0.781·59-s + 0.768·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.62034046341440, −14.99907296369228, −14.32470688132399, −14.06897326605110, −13.33187862241255, −13.07969860538222, −12.42022840280306, −11.94479932105054, −11.11691718180344, −10.67643612185310, −9.929058291222910, −9.614711788353702, −8.993774814558035, −8.584188337792284, −7.760126052980759, −7.290325401891593, −6.560009794051454, −6.166877889910977, −5.324610273185608, −4.818595381021835, −3.870804169455698, −3.335166083914896, −2.776036595443703, −1.923594591935979, −1.195162785633900, 0,
1.195162785633900, 1.923594591935979, 2.776036595443703, 3.335166083914896, 3.870804169455698, 4.818595381021835, 5.324610273185608, 6.166877889910977, 6.560009794051454, 7.290325401891593, 7.760126052980759, 8.584188337792284, 8.993774814558035, 9.614711788353702, 9.929058291222910, 10.67643612185310, 11.11691718180344, 11.94479932105054, 12.42022840280306, 13.07969860538222, 13.33187862241255, 14.06897326605110, 14.32470688132399, 14.99907296369228, 15.62034046341440
