| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s − 7-s + 8·10-s + 2·13-s + 2·14-s − 4·16-s + 4·17-s + 3·19-s − 8·20-s − 2·23-s + 11·25-s − 4·26-s − 2·28-s + 6·29-s − 5·31-s + 8·32-s − 8·34-s + 4·35-s + 3·37-s − 6·38-s − 2·41-s − 12·43-s + 4·46-s − 2·47-s − 6·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.78·5-s − 0.377·7-s + 2.52·10-s + 0.554·13-s + 0.534·14-s − 16-s + 0.970·17-s + 0.688·19-s − 1.78·20-s − 0.417·23-s + 11/5·25-s − 0.784·26-s − 0.377·28-s + 1.11·29-s − 0.898·31-s + 1.41·32-s − 1.37·34-s + 0.676·35-s + 0.493·37-s − 0.973·38-s − 0.312·41-s − 1.82·43-s + 0.589·46-s − 0.291·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−9.427976724918087578878267878502, −8.345816310960391722598141882781, −8.118926450037046023972998230501, −7.28899889706555778737663428863, −6.56051958738602663681229923169, −5.05774622025620960162902603755, −3.93444342612617848308073041065, −3.09166168739687900450908810991, −1.23603098910894570836676745841, 0,
1.23603098910894570836676745841, 3.09166168739687900450908810991, 3.93444342612617848308073041065, 5.05774622025620960162902603755, 6.56051958738602663681229923169, 7.28899889706555778737663428863, 8.118926450037046023972998230501, 8.345816310960391722598141882781, 9.427976724918087578878267878502
