| L(s) = 1 | − 3-s + 2·5-s + 7-s + 9-s + 4·11-s − 2·13-s − 2·15-s − 4·19-s − 21-s + 8·23-s − 25-s − 27-s − 6·29-s − 4·33-s + 2·35-s + 2·37-s + 2·39-s − 10·41-s + 4·43-s + 2·45-s + 49-s + 6·53-s + 8·55-s + 4·57-s + 4·59-s − 6·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.516·15-s − 0.917·19-s − 0.218·21-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.696·33-s + 0.338·35-s + 0.328·37-s + 0.320·39-s − 1.56·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-s + 0.824·53-s + 1.07·55-s + 0.529·57-s + 0.520·59-s − 0.768·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.96094759713748, −13.53633425835602, −13.02566594627195, −12.60588457785884, −12.00960885402547, −11.58277316996603, −11.09270210982521, −10.62628560940800, −10.13254796476960, −9.446142655941596, −9.258518654037324, −8.677922489915759, −8.033106914569330, −7.333386567383832, −6.813265698719520, −6.480613768935399, −5.878843725092477, −5.221422522485305, −5.013937345604059, −4.109806171164374, −3.798544491260758, −2.830543113126808, −2.176323265900228, −1.577002740683889, −1.015155294290434, 0,
1.015155294290434, 1.577002740683889, 2.176323265900228, 2.830543113126808, 3.798544491260758, 4.109806171164374, 5.013937345604059, 5.221422522485305, 5.878843725092477, 6.480613768935399, 6.813265698719520, 7.333386567383832, 8.033106914569330, 8.677922489915759, 9.258518654037324, 9.446142655941596, 10.13254796476960, 10.62628560940800, 11.09270210982521, 11.58277316996603, 12.00960885402547, 12.60588457785884, 13.02566594627195, 13.53633425835602, 13.96094759713748
