| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 3·7-s − 8-s + 9-s − 4·11-s − 12-s + 13-s − 3·14-s + 16-s − 18-s − 7·19-s − 3·21-s + 4·22-s − 6·23-s + 24-s − 26-s − 27-s + 3·28-s − 29-s − 7·31-s − 32-s + 4·33-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 0.277·13-s − 0.801·14-s + 1/4·16-s − 0.235·18-s − 1.60·19-s − 0.654·21-s + 0.852·22-s − 1.25·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.566·28-s − 0.185·29-s − 1.25·31-s − 0.176·32-s + 0.696·33-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 43 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−12.85966858483068, −12.37958031387084, −12.09052946899095, −11.40612266491265, −10.98361613057671, −10.72734809381315, −10.48153545863732, −9.693571363970518, −9.453108035923910, −8.637127019974276, −8.281799486850097, −7.939489036203137, −7.555873564374111, −6.952412188739015, −6.260068849656997, −6.081739087506953, −5.308633666650702, −4.982684702189854, −4.450215144122435, −3.817291749418041, −3.236680214430247, −2.282431730704807, −2.003117818907265, −1.538121628244379, −0.5748377869413531, 0,
0.5748377869413531, 1.538121628244379, 2.003117818907265, 2.282431730704807, 3.236680214430247, 3.817291749418041, 4.450215144122435, 4.982684702189854, 5.308633666650702, 6.081739087506953, 6.260068849656997, 6.952412188739015, 7.555873564374111, 7.939489036203137, 8.281799486850097, 8.637127019974276, 9.453108035923910, 9.693571363970518, 10.48153545863732, 10.72734809381315, 10.98361613057671, 11.40612266491265, 12.09052946899095, 12.37958031387084, 12.85966858483068
