| L(s) = 1 | + 2·5-s + 2·11-s + 4·13-s − 19-s + 8·23-s − 25-s + 2·29-s − 2·31-s − 8·37-s − 2·41-s − 4·43-s + 4·47-s − 2·53-s + 4·55-s + 10·61-s + 8·65-s − 16·71-s − 6·73-s − 14·79-s + 6·83-s − 18·89-s − 2·95-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.603·11-s + 1.10·13-s − 0.229·19-s + 1.66·23-s − 1/5·25-s + 0.371·29-s − 0.359·31-s − 1.31·37-s − 0.312·41-s − 0.609·43-s + 0.583·47-s − 0.274·53-s + 0.539·55-s + 1.28·61-s + 0.992·65-s − 1.89·71-s − 0.702·73-s − 1.57·79-s + 0.658·83-s − 1.90·89-s − 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.62843251107758, −13.29768693696324, −12.82855672737989, −12.35637532317341, −11.63021727635211, −11.34041225006105, −10.81345653105032, −10.16623535790080, −10.01788285212924, −9.183629336104209, −8.796691212611317, −8.634772729224670, −7.825297758097768, −7.097709123521595, −6.793062234238126, −6.246005186232627, −5.685631494415036, −5.327009512927898, −4.625885118836413, −4.035297194286163, −3.407433463092674, −2.918828623809754, −2.148062294342513, −1.454301327511861, −1.128577709722861, 0,
1.128577709722861, 1.454301327511861, 2.148062294342513, 2.918828623809754, 3.407433463092674, 4.035297194286163, 4.625885118836413, 5.327009512927898, 5.685631494415036, 6.246005186232627, 6.793062234238126, 7.097709123521595, 7.825297758097768, 8.634772729224670, 8.796691212611317, 9.183629336104209, 10.01788285212924, 10.16623535790080, 10.81345653105032, 11.34041225006105, 11.63021727635211, 12.35637532317341, 12.82855672737989, 13.29768693696324, 13.62843251107758
