| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s + 9-s + 10-s − 2·11-s − 12-s + 2·14-s + 15-s + 16-s + 7·17-s − 18-s − 7·19-s − 20-s + 2·21-s + 2·22-s − 2·23-s + 24-s − 4·25-s − 27-s − 2·28-s − 8·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s − 1.60·19-s − 0.223·20-s + 0.436·21-s + 0.426·22-s − 0.417·23-s + 0.204·24-s − 4/5·25-s − 0.192·27-s − 0.377·28-s − 1.48·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1633202294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1633202294\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.84833216682583, −14.48110731392389, −13.46095375019472, −13.10314873175682, −12.54564027119504, −12.13522756692083, −11.44223421689255, −11.19136502548865, −10.34194889085475, −10.06138935268847, −9.652893688596492, −8.961826658155873, −8.178536847297208, −7.839731145402296, −7.390313301492558, −6.585293759818997, −6.077316216256639, −5.729190679700250, −4.873048515595492, −4.161334771397671, −3.508127834041301, −2.872869399024357, −2.036691555810930, −1.238770072428164, −0.1795044961208321,
0.1795044961208321, 1.238770072428164, 2.036691555810930, 2.872869399024357, 3.508127834041301, 4.161334771397671, 4.873048515595492, 5.729190679700250, 6.077316216256639, 6.585293759818997, 7.390313301492558, 7.839731145402296, 8.178536847297208, 8.961826658155873, 9.652893688596492, 10.06138935268847, 10.34194889085475, 11.19136502548865, 11.44223421689255, 12.13522756692083, 12.54564027119504, 13.10314873175682, 13.46095375019472, 14.48110731392389, 14.84833216682583
