| L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s + 11-s + 2·13-s − 2·15-s + 17-s − 6·19-s + 21-s − 6·23-s − 25-s + 27-s − 29-s − 4·31-s + 33-s − 2·35-s + 2·39-s − 9·41-s + 4·43-s − 2·45-s + 3·47-s − 6·49-s + 51-s + 11·53-s − 2·55-s − 6·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.516·15-s + 0.242·17-s − 1.37·19-s + 0.218·21-s − 1.25·23-s − 1/5·25-s + 0.192·27-s − 0.185·29-s − 0.718·31-s + 0.174·33-s − 0.338·35-s + 0.320·39-s − 1.40·41-s + 0.609·43-s − 0.298·45-s + 0.437·47-s − 6/7·49-s + 0.140·51-s + 1.51·53-s − 0.269·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.008813593379173988420335705510, −7.48491470873779780220550534418, −6.59084177723032786182916240672, −5.86776283215741156445806732941, −4.80738272202713631953127570851, −3.94522715787654390236984668070, −3.65062157764984085911544906432, −2.39620114647415447973826006313, −1.51687879531338122694181846797, 0,
1.51687879531338122694181846797, 2.39620114647415447973826006313, 3.65062157764984085911544906432, 3.94522715787654390236984668070, 4.80738272202713631953127570851, 5.86776283215741156445806732941, 6.59084177723032786182916240672, 7.48491470873779780220550534418, 8.008813593379173988420335705510
