| L(s) = 1 | − 2-s + 2·3-s + 4-s + 5-s − 2·6-s − 8-s + 9-s − 10-s + 2·12-s + 2·15-s + 16-s + 6·17-s − 18-s + 20-s − 6·23-s − 2·24-s + 25-s − 4·27-s − 2·30-s − 4·31-s − 32-s − 6·34-s + 36-s + 2·37-s − 40-s − 12·41-s − 12·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.223·20-s − 1.25·23-s − 0.408·24-s + 1/5·25-s − 0.769·27-s − 0.365·30-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.328·37-s − 0.158·40-s − 1.87·41-s − 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 204490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 204490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.033779513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.033779513\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.05608513482936, −12.70007942806720, −11.92346020783256, −11.70241669214719, −11.19331784746813, −10.29316282991440, −10.17279334045137, −9.644017353406697, −9.418423653566534, −8.528267753330334, −8.448083557852652, −8.006913300055641, −7.451675618450849, −6.991893435923325, −6.352414570671511, −5.877515704274307, −5.285834813244807, −4.811897862782977, −3.808284571871025, −3.471257607217245, −3.051160337636401, −2.330800665902119, −1.777210813536590, −1.437780860384723, −0.3909117331680548,
0.3909117331680548, 1.437780860384723, 1.777210813536590, 2.330800665902119, 3.051160337636401, 3.471257607217245, 3.808284571871025, 4.811897862782977, 5.285834813244807, 5.877515704274307, 6.352414570671511, 6.991893435923325, 7.451675618450849, 8.006913300055641, 8.448083557852652, 8.528267753330334, 9.418423653566534, 9.644017353406697, 10.17279334045137, 10.29316282991440, 11.19331784746813, 11.70241669214719, 11.92346020783256, 12.70007942806720, 13.05608513482936
