| L(s) = 1 | − 5-s + 7-s − 11-s − 2·13-s − 6·17-s + 4·19-s + 25-s + 6·29-s + 8·31-s − 35-s + 10·37-s + 6·41-s − 8·43-s + 49-s + 6·53-s + 55-s + 12·59-s − 2·61-s + 2·65-s + 4·67-s − 12·71-s − 10·73-s − 77-s − 4·79-s − 12·83-s + 6·85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.301·11-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.169·35-s + 1.64·37-s + 0.937·41-s − 1.21·43-s + 1/7·49-s + 0.824·53-s + 0.134·55-s + 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s − 1.42·71-s − 1.17·73-s − 0.113·77-s − 0.450·79-s − 1.31·83-s + 0.650·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 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.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.19732370241921, −12.80055680020878, −12.14877497378834, −11.71407667872878, −11.44380214371501, −10.99490693381445, −10.33595508430700, −9.942863569490811, −9.573752978133845, −8.747165307251894, −8.573939653963473, −7.993272792265067, −7.544974709610078, −6.989728683427247, −6.632968648292839, −5.980921135174466, −5.436389616893724, −4.843244947124691, −4.319356135755251, −4.181793886873522, −3.142520705084073, −2.694080657725240, −2.317656648656603, −1.371011942834884, −0.8033516317495055, 0,
0.8033516317495055, 1.371011942834884, 2.317656648656603, 2.694080657725240, 3.142520705084073, 4.181793886873522, 4.319356135755251, 4.843244947124691, 5.436389616893724, 5.980921135174466, 6.632968648292839, 6.989728683427247, 7.544974709610078, 7.993272792265067, 8.573939653963473, 8.747165307251894, 9.573752978133845, 9.942863569490811, 10.33595508430700, 10.99490693381445, 11.44380214371501, 11.71407667872878, 12.14877497378834, 12.80055680020878, 13.19732370241921
