| L(s) = 1 | + 3-s − 2·4-s + 3·5-s + 7-s + 9-s + 6·11-s − 2·12-s − 13-s + 3·15-s + 4·16-s − 4·19-s − 6·20-s + 21-s + 3·23-s + 4·25-s + 27-s − 2·28-s + 3·29-s + 5·31-s + 6·33-s + 3·35-s − 2·36-s − 7·37-s − 39-s + 6·41-s − 10·43-s − 12·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1.34·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.577·12-s − 0.277·13-s + 0.774·15-s + 16-s − 0.917·19-s − 1.34·20-s + 0.218·21-s + 0.625·23-s + 4/5·25-s + 0.192·27-s − 0.377·28-s + 0.557·29-s + 0.898·31-s + 1.04·33-s + 0.507·35-s − 1/3·36-s − 1.15·37-s − 0.160·39-s + 0.937·41-s − 1.52·43-s − 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.739775183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.739775183\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 127 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−8.773764419467853775124703867780, −8.583184694018683567028279973443, −7.31727170843408812858724922870, −6.47128817451613464407808826900, −5.79892141358391166239107819835, −4.77844105399236766130837684247, −4.19576988986652335266289856464, −3.16950955287184962161065121707, −1.97142810612244721976820075895, −1.13352671817283173196288281982,
1.13352671817283173196288281982, 1.97142810612244721976820075895, 3.16950955287184962161065121707, 4.19576988986652335266289856464, 4.77844105399236766130837684247, 5.79892141358391166239107819835, 6.47128817451613464407808826900, 7.31727170843408812858724922870, 8.583184694018683567028279973443, 8.773764419467853775124703867780
