| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 11-s − 12-s + 2·13-s + 16-s − 4·17-s − 18-s − 22-s + 2·23-s + 24-s − 2·26-s − 27-s − 8·31-s − 32-s − 33-s + 4·34-s + 36-s + 12·37-s − 2·39-s + 2·41-s + 12·43-s + 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 0.554·13-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.213·22-s + 0.417·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 1.43·31-s − 0.176·32-s − 0.174·33-s + 0.685·34-s + 1/6·36-s + 1.97·37-s − 0.320·39-s + 0.312·41-s + 1.82·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 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
−14.33186533155953, −13.64795096399465, −13.15476847129290, −12.59749694024002, −12.33169507356871, −11.38499962888793, −11.12893598878645, −11.04061035016777, −10.16179741257788, −9.776060200247150, −9.054584379580188, −8.895828559644641, −8.198852619056701, −7.520172189953412, −7.138871156046383, −6.598649451729536, −5.937103206445750, −5.680640982155973, −4.873125706159210, −4.092707308724514, −3.856646738074220, −2.747809765912766, −2.333861955090093, −1.414729067894990, −0.8736469948349985, 0,
0.8736469948349985, 1.414729067894990, 2.333861955090093, 2.747809765912766, 3.856646738074220, 4.092707308724514, 4.873125706159210, 5.680640982155973, 5.937103206445750, 6.598649451729536, 7.138871156046383, 7.520172189953412, 8.198852619056701, 8.895828559644641, 9.054584379580188, 9.776060200247150, 10.16179741257788, 11.04061035016777, 11.12893598878645, 11.38499962888793, 12.33169507356871, 12.59749694024002, 13.15476847129290, 13.64795096399465, 14.33186533155953
