| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 11-s − 12-s + 4·13-s + 16-s + 6·17-s − 18-s − 2·19-s − 22-s + 24-s − 4·26-s − 27-s − 6·29-s − 2·31-s − 32-s − 33-s − 6·34-s + 36-s − 2·37-s + 2·38-s − 4·39-s + 2·41-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 + 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.213·22-s + 0.204·24-s − 0.784·26-s − 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.176·32-s − 0.174·33-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.324·38-s − 0.640·39-s + 0.312·41-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 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−14.31241055166681, −13.74565978938173, −13.12503711000355, −12.58251794665456, −12.23431308305895, −11.64447934766798, −11.13746717454448, −10.67899561451134, −10.44484204735785, −9.477341720570812, −9.407099954376148, −8.777280345707986, −7.970035059830658, −7.778646476145874, −7.159479321024916, −6.400698439682184, −6.106769571368539, −5.577814861695772, −4.965496651368183, −4.156283484453883, −3.585695758493142, −3.081712359017111, −2.096684644923281, −1.454605448247736, −0.9164567270914665, 0,
0.9164567270914665, 1.454605448247736, 2.096684644923281, 3.081712359017111, 3.585695758493142, 4.156283484453883, 4.965496651368183, 5.577814861695772, 6.106769571368539, 6.400698439682184, 7.159479321024916, 7.778646476145874, 7.970035059830658, 8.777280345707986, 9.407099954376148, 9.477341720570812, 10.44484204735785, 10.67899561451134, 11.13746717454448, 11.64447934766798, 12.23431308305895, 12.58251794665456, 13.12503711000355, 13.74565978938173, 14.31241055166681
