| L(s) = 1 | − 2·7-s + 4·11-s − 13-s + 3·17-s + 19-s − 23-s − 5·25-s + 2·29-s + 3·31-s − 5·37-s − 7·41-s − 3·43-s + 2·47-s − 3·49-s + 6·53-s − 59-s − 5·61-s + 3·67-s + 12·71-s − 8·77-s − 10·79-s − 18·83-s − 10·89-s + 2·91-s + 13·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.20·11-s − 0.277·13-s + 0.727·17-s + 0.229·19-s − 0.208·23-s − 25-s + 0.371·29-s + 0.538·31-s − 0.821·37-s − 1.09·41-s − 0.457·43-s + 0.291·47-s − 3/7·49-s + 0.824·53-s − 0.130·59-s − 0.640·61-s + 0.366·67-s + 1.42·71-s − 0.911·77-s − 1.12·79-s − 1.97·83-s − 1.05·89-s + 0.209·91-s + 1.31·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−16.00259818598353, −15.62299367642131, −15.03754189353076, −14.33495799596883, −13.95059003946614, −13.44466375653400, −12.68248915555421, −12.19165836983747, −11.74317591826831, −11.24820761899399, −10.24189194258763, −9.952163799360214, −9.457224742924001, −8.729861201577434, −8.226055218449996, −7.399035993629699, −6.861946589415342, −6.295268232089793, −5.702085480924754, −4.964314153811326, −4.123842322889729, −3.554452735949484, −2.943542497012699, −1.932291259248875, −1.130974961373579, 0,
1.130974961373579, 1.932291259248875, 2.943542497012699, 3.554452735949484, 4.123842322889729, 4.964314153811326, 5.702085480924754, 6.295268232089793, 6.861946589415342, 7.399035993629699, 8.226055218449996, 8.729861201577434, 9.457224742924001, 9.952163799360214, 10.24189194258763, 11.24820761899399, 11.74317591826831, 12.19165836983747, 12.68248915555421, 13.44466375653400, 13.95059003946614, 14.33495799596883, 15.03754189353076, 15.62299367642131, 16.00259818598353
