| L(s) = 1 | + 7-s − 3·9-s + 2·13-s − 17-s − 4·19-s − 6·29-s + 6·37-s − 6·41-s − 12·43-s + 8·47-s + 49-s + 2·53-s − 4·59-s + 2·61-s − 3·63-s + 12·67-s − 2·73-s + 8·79-s + 9·81-s + 12·83-s + 10·89-s + 2·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 0.554·13-s − 0.242·17-s − 0.917·19-s − 1.11·29-s + 0.986·37-s − 0.937·41-s − 1.82·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s − 0.520·59-s + 0.256·61-s − 0.377·63-s + 1.46·67-s − 0.234·73-s + 0.900·79-s + 81-s + 1.31·83-s + 1.05·89-s + 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.88793360790700, −14.38933926872512, −13.75244246146460, −13.35780744085018, −12.87960679870942, −12.19167129274189, −11.63367009122284, −11.27972929838871, −10.76342786619336, −10.27811505933038, −9.563043400757523, −8.912137572016022, −8.627292472458330, −7.999441459513032, −7.570194970493450, −6.667527053752920, −6.326237005404367, −5.663367329420181, −5.123488046724930, −4.518986543965208, −3.712357044851454, −3.317133582594801, −2.328653629368799, −1.949811531193886, −0.8991224090459984, 0,
0.8991224090459984, 1.949811531193886, 2.328653629368799, 3.317133582594801, 3.712357044851454, 4.518986543965208, 5.123488046724930, 5.663367329420181, 6.326237005404367, 6.667527053752920, 7.570194970493450, 7.999441459513032, 8.627292472458330, 8.912137572016022, 9.563043400757523, 10.27811505933038, 10.76342786619336, 11.27972929838871, 11.63367009122284, 12.19167129274189, 12.87960679870942, 13.35780744085018, 13.75244246146460, 14.38933926872512, 14.88793360790700
