| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s − 3·9-s + 10-s − 4·11-s − 14-s + 16-s + 2·17-s − 3·18-s + 4·19-s + 20-s − 4·22-s + 8·23-s + 25-s − 28-s − 2·29-s − 4·31-s + 32-s + 2·34-s − 35-s − 3·36-s − 10·37-s + 4·38-s + 40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s − 9-s + 0.316·10-s − 1.20·11-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s + 0.917·19-s + 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.188·28-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.342·34-s − 0.169·35-s − 1/2·36-s − 1.64·37-s + 0.648·38-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−16.74455605944361, −16.01126539054033, −15.50744894446311, −14.92581646065956, −14.29777159122912, −13.87171216741512, −13.25446980335980, −12.79469451088088, −12.28044695072582, −11.46652175715745, −11.02041889416768, −10.43835072451674, −9.785357828414011, −9.065771973502920, −8.521958566749765, −7.582181235757034, −7.238683642845164, −6.343468676597367, −5.639941793168100, −5.290594693802244, −4.697929003641665, −3.413277378749485, −3.102126560991714, −2.396487371636783, −1.327449998245200, 0,
1.327449998245200, 2.396487371636783, 3.102126560991714, 3.413277378749485, 4.697929003641665, 5.290594693802244, 5.639941793168100, 6.343468676597367, 7.238683642845164, 7.582181235757034, 8.521958566749765, 9.065771973502920, 9.785357828414011, 10.43835072451674, 11.02041889416768, 11.46652175715745, 12.28044695072582, 12.79469451088088, 13.25446980335980, 13.87171216741512, 14.29777159122912, 14.92581646065956, 15.50744894446311, 16.01126539054033, 16.74455605944361
