| L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s − 11-s − 2·13-s + 16-s + 2·17-s + 3·18-s + 4·19-s + 22-s − 6·23-s + 2·26-s + 9·29-s − 6·31-s − 32-s − 2·34-s − 3·36-s + 9·37-s − 4·38-s + 2·41-s − 10·43-s − 44-s + 6·46-s + 3·47-s − 2·52-s + 7·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s − 0.301·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.707·18-s + 0.917·19-s + 0.213·22-s − 1.25·23-s + 0.392·26-s + 1.67·29-s − 1.07·31-s − 0.176·32-s − 0.342·34-s − 1/2·36-s + 1.47·37-s − 0.648·38-s + 0.312·41-s − 1.52·43-s − 0.150·44-s + 0.884·46-s + 0.437·47-s − 0.277·52-s + 0.961·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.616340975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.616340975\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 7 T + p T^{2} \) | 1.53.ah |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 11 T + p T^{2} \) | 1.71.al |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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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
−13.02521372990181, −12.46517513018880, −12.00316274068065, −11.71430518765933, −11.21863536056257, −10.68346809651751, −10.20682054308606, −9.712207070269691, −9.441522694532669, −8.775150218847319, −8.298611269741251, −7.815508927151366, −7.646184234062210, −6.785005779332442, −6.434105755320592, −5.824694788142750, −5.313481051409445, −4.946589096893838, −4.111144977728276, −3.454017182404589, −2.992880777547968, −2.334400089908396, −1.955734850628265, −0.8945362241135673, −0.5055413472506377,
0.5055413472506377, 0.8945362241135673, 1.955734850628265, 2.334400089908396, 2.992880777547968, 3.454017182404589, 4.111144977728276, 4.946589096893838, 5.313481051409445, 5.824694788142750, 6.434105755320592, 6.785005779332442, 7.646184234062210, 7.815508927151366, 8.298611269741251, 8.775150218847319, 9.441522694532669, 9.712207070269691, 10.20682054308606, 10.68346809651751, 11.21863536056257, 11.71430518765933, 12.00316274068065, 12.46517513018880, 13.02521372990181
