| L(s) = 1 | + 5-s − 3·9-s − 11-s − 2·13-s − 2·17-s + 4·19-s + 25-s − 6·29-s − 4·31-s − 10·37-s − 6·41-s − 4·43-s − 3·45-s + 8·47-s − 7·49-s + 6·53-s − 55-s − 12·59-s + 2·61-s − 2·65-s − 8·67-s + 12·71-s − 2·73-s + 8·79-s + 9·81-s + 12·83-s − 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s − 0.301·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 1.64·37-s − 0.937·41-s − 0.609·43-s − 0.447·45-s + 1.16·47-s − 49-s + 0.824·53-s − 0.134·55-s − 1.56·59-s + 0.256·61-s − 0.248·65-s − 0.977·67-s + 1.42·71-s − 0.234·73-s + 0.900·79-s + 81-s + 1.31·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 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 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.018259819344327273950973449663, −8.153842154433902963123081145889, −7.32038477439307062853641022538, −6.50478262992673508196850138672, −5.46692176962337557138711266920, −5.10594091471141189642712861975, −3.71439352036814173065774275481, −2.80574037349613025778059374744, −1.78206243921462941212360072602, 0,
1.78206243921462941212360072602, 2.80574037349613025778059374744, 3.71439352036814173065774275481, 5.10594091471141189642712861975, 5.46692176962337557138711266920, 6.50478262992673508196850138672, 7.32038477439307062853641022538, 8.153842154433902963123081145889, 9.018259819344327273950973449663
