| L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s + 7-s − 8-s − 2·9-s + 3·10-s − 6·11-s − 12-s − 13-s − 14-s + 3·15-s + 16-s + 3·17-s + 2·18-s − 2·19-s − 3·20-s − 21-s + 6·22-s + 24-s + 4·25-s + 26-s + 5·27-s + 28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s − 1.80·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.458·19-s − 0.670·20-s − 0.218·21-s + 1.27·22-s + 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.962·27-s + 0.188·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 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 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.94637746633941, −14.74543276798212, −13.99867412582348, −13.21103960324332, −12.67844317797090, −12.22481558933920, −11.68103063743456, −11.23309300781103, −10.85959800257571, −10.35763604300998, −9.866248595082933, −8.916199446936589, −8.558045837754044, −7.908706498954856, −7.616068232342124, −7.230350744671101, −6.322162753236313, −5.710715764479344, −5.082063579228493, −4.765685283373647, −3.675440218912180, −3.242025696322316, −2.473880799548411, −1.699782490245100, −0.5014307089860055, 0,
0.5014307089860055, 1.699782490245100, 2.473880799548411, 3.242025696322316, 3.675440218912180, 4.765685283373647, 5.082063579228493, 5.710715764479344, 6.322162753236313, 7.230350744671101, 7.616068232342124, 7.908706498954856, 8.558045837754044, 8.916199446936589, 9.866248595082933, 10.35763604300998, 10.85959800257571, 11.23309300781103, 11.68103063743456, 12.22481558933920, 12.67844317797090, 13.21103960324332, 13.99867412582348, 14.74543276798212, 14.94637746633941
