| L(s) = 1 | − 4·7-s + 2·11-s + 13-s − 5·17-s − 3·19-s + 3·23-s − 4·29-s − 3·31-s − 10·37-s − 8·41-s − 4·43-s + 9·49-s − 5·53-s + 2·59-s + 5·61-s − 12·67-s − 10·71-s + 8·73-s − 8·77-s + 79-s + 3·83-s + 10·89-s − 4·91-s + 4·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.603·11-s + 0.277·13-s − 1.21·17-s − 0.688·19-s + 0.625·23-s − 0.742·29-s − 0.538·31-s − 1.64·37-s − 1.24·41-s − 0.609·43-s + 9/7·49-s − 0.686·53-s + 0.260·59-s + 0.640·61-s − 1.46·67-s − 1.18·71-s + 0.936·73-s − 0.911·77-s + 0.112·79-s + 0.329·83-s + 1.05·89-s − 0.419·91-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.61533879909870, −13.30890224815055, −13.13135022264535, −12.40687781323925, −12.13018732804821, −11.45729230061096, −11.02056255777627, −10.41651115048374, −10.14979354366617, −9.394889315764832, −9.066618533116038, −8.751428346942998, −8.153174649474549, −7.301681270086712, −6.921016229220898, −6.487465135067286, −6.182149267830398, −5.441328139005545, −4.885141372326244, −4.221108840669591, −3.520501254266757, −3.417098327260610, −2.541665027206013, −1.931218787852841, −1.247800966580213, 0, 0,
1.247800966580213, 1.931218787852841, 2.541665027206013, 3.417098327260610, 3.520501254266757, 4.221108840669591, 4.885141372326244, 5.441328139005545, 6.182149267830398, 6.487465135067286, 6.921016229220898, 7.301681270086712, 8.153174649474549, 8.751428346942998, 9.066618533116038, 9.394889315764832, 10.14979354366617, 10.41651115048374, 11.02056255777627, 11.45729230061096, 12.13018732804821, 12.40687781323925, 13.13135022264535, 13.30890224815055, 13.61533879909870
