| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 4·7-s − 8-s + 9-s − 10-s − 4·11-s − 12-s − 2·13-s + 4·14-s − 15-s + 16-s − 6·17-s − 18-s + 4·19-s + 20-s + 4·21-s + 4·22-s + 24-s + 25-s + 2·26-s − 27-s − 4·28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.554·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.872·21-s + 0.852·22-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.755·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 349 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 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.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.98367011941774, −16.05443201876604, −15.90409224146280, −15.63669133031074, −14.71994941801476, −13.67374669329873, −13.48098388531486, −12.69741054766908, −12.30424004215298, −11.59988242277830, −10.84197222843173, −10.26278733980183, −9.941799132169362, −9.357173686172284, −8.679875451182239, −7.942372799015750, −7.100824137754904, −6.656380114330430, −6.158987017211467, −5.308379616091107, −4.755140640513006, −3.606304088945999, −2.725117986795152, −2.271276963930499, −0.8503732929138383, 0,
0.8503732929138383, 2.271276963930499, 2.725117986795152, 3.606304088945999, 4.755140640513006, 5.308379616091107, 6.158987017211467, 6.656380114330430, 7.100824137754904, 7.942372799015750, 8.679875451182239, 9.357173686172284, 9.941799132169362, 10.26278733980183, 10.84197222843173, 11.59988242277830, 12.30424004215298, 12.69741054766908, 13.48098388531486, 13.67374669329873, 14.71994941801476, 15.63669133031074, 15.90409224146280, 16.05443201876604, 16.98367011941774
