| L(s) = 1 | + 2·7-s + 2·11-s − 6·13-s − 2·17-s + 4·23-s − 8·31-s − 2·37-s − 2·41-s + 4·43-s + 8·47-s − 3·49-s + 6·53-s + 10·59-s − 2·61-s + 8·67-s − 12·71-s − 4·73-s + 4·77-s − 4·83-s + 10·89-s − 12·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.603·11-s − 1.66·13-s − 0.485·17-s + 0.834·23-s − 1.43·31-s − 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.824·53-s + 1.30·59-s − 0.256·61-s + 0.977·67-s − 1.42·71-s − 0.468·73-s + 0.455·77-s − 0.439·83-s + 1.05·89-s − 1.25·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−16.43991477160014, −15.86869673127612, −14.97308658029380, −14.79429693834879, −14.35426596759064, −13.62677543993374, −13.01645150911515, −12.35079518511054, −11.95214411822746, −11.27927074791834, −10.81637053847256, −10.11273870237079, −9.465733616383356, −8.941735381642752, −8.381370730060747, −7.429663162539531, −7.248355247160462, −6.512708308604890, −5.518835883608753, −5.114489591072073, −4.392730218924860, −3.761023968233590, −2.717216422884158, −2.107613467000151, −1.206742240870161, 0,
1.206742240870161, 2.107613467000151, 2.717216422884158, 3.761023968233590, 4.392730218924860, 5.114489591072073, 5.518835883608753, 6.512708308604890, 7.248355247160462, 7.429663162539531, 8.381370730060747, 8.941735381642752, 9.465733616383356, 10.11273870237079, 10.81637053847256, 11.27927074791834, 11.95214411822746, 12.35079518511054, 13.01645150911515, 13.62677543993374, 14.35426596759064, 14.79429693834879, 14.97308658029380, 15.86869673127612, 16.43991477160014
