| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 6·11-s + 12-s − 4·13-s + 14-s + 16-s − 18-s − 21-s − 6·22-s − 6·23-s − 24-s + 4·26-s + 27-s − 28-s − 6·29-s + 4·31-s − 32-s + 6·33-s + 36-s + 2·37-s − 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.218·21-s − 1.27·22-s − 1.25·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.04·33-s + 1/6·36-s + 0.328·37-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−12.72740289165912, −11.98050090333375, −11.94923585912594, −11.34682666336136, −10.94867870438336, −10.15994342672403, −9.782054660693847, −9.591133090011101, −9.246311209074871, −8.577591118116986, −8.201992022128070, −7.836097860577141, −7.110512626339412, −6.856873569703132, −6.449748600791100, −5.876979713957488, −5.326249372611103, −4.588782782261376, −4.081475173059456, −3.657767246183656, −3.145397490871961, −2.417827182273511, −2.001594512287872, −1.451820553674899, −0.7567663233185626, 0,
0.7567663233185626, 1.451820553674899, 2.001594512287872, 2.417827182273511, 3.145397490871961, 3.657767246183656, 4.081475173059456, 4.588782782261376, 5.326249372611103, 5.876979713957488, 6.449748600791100, 6.856873569703132, 7.110512626339412, 7.836097860577141, 8.201992022128070, 8.577591118116986, 9.246311209074871, 9.591133090011101, 9.782054660693847, 10.15994342672403, 10.94867870438336, 11.34682666336136, 11.94923585912594, 11.98050090333375, 12.72740289165912
