| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 2·5-s − 2·6-s + 9-s + 4·10-s + 2·11-s − 2·12-s + 13-s − 2·15-s − 4·16-s + 2·18-s + 19-s + 4·20-s + 4·22-s − 25-s + 2·26-s − 27-s − 4·29-s − 4·30-s − 9·31-s − 8·32-s − 2·33-s + 2·36-s − 3·37-s + 2·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 0.894·5-s − 0.816·6-s + 1/3·9-s + 1.26·10-s + 0.603·11-s − 0.577·12-s + 0.277·13-s − 0.516·15-s − 16-s + 0.471·18-s + 0.229·19-s + 0.894·20-s + 0.852·22-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.742·29-s − 0.730·30-s − 1.61·31-s − 1.41·32-s − 0.348·33-s + 1/3·36-s − 0.493·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.90596524176067, −14.25716280006158, −13.97478508810462, −13.33354916790157, −13.05899049541961, −12.43458436313728, −12.04986268469637, −11.48020805784338, −10.89810968854673, −10.56733926887736, −9.663035601116145, −9.192045806660466, −8.926631301402713, −7.736892858822922, −7.318339954369691, −6.518883668584963, −6.124899615210211, −5.703672587482848, −5.206582894036081, −4.622388650742536, −3.849353851605251, −3.563092089552345, −2.582166602995999, −1.967458139789342, −1.228574797926795, 0,
1.228574797926795, 1.967458139789342, 2.582166602995999, 3.563092089552345, 3.849353851605251, 4.622388650742536, 5.206582894036081, 5.703672587482848, 6.124899615210211, 6.518883668584963, 7.318339954369691, 7.736892858822922, 8.926631301402713, 9.192045806660466, 9.663035601116145, 10.56733926887736, 10.89810968854673, 11.48020805784338, 12.04986268469637, 12.43458436313728, 13.05899049541961, 13.33354916790157, 13.97478508810462, 14.25716280006158, 14.90596524176067
