| L(s) = 1 | − 3-s − 2·7-s + 9-s + 13-s − 2·19-s + 2·21-s − 8·23-s − 5·25-s − 27-s − 6·29-s − 2·31-s + 6·37-s − 39-s + 4·43-s + 8·47-s − 3·49-s − 6·53-s + 2·57-s + 4·59-s − 2·61-s − 2·63-s + 2·67-s + 8·69-s − 4·71-s + 2·73-s + 5·75-s − 12·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.458·19-s + 0.436·21-s − 1.66·23-s − 25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.986·37-s − 0.160·39-s + 0.609·43-s + 1.16·47-s − 3/7·49-s − 0.824·53-s + 0.264·57-s + 0.520·59-s − 0.256·61-s − 0.251·63-s + 0.244·67-s + 0.963·69-s − 0.474·71-s + 0.234·73-s + 0.577·75-s − 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 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.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−12.87734060775100, −12.25188516049543, −11.86866205673183, −11.30445637789705, −11.12963666617640, −10.29551819630084, −10.18748667622353, −9.613108014566733, −9.216069971700191, −8.730730078912845, −8.061238059320114, −7.655769416543019, −7.275620583337451, −6.618133333841738, −6.090189001947663, −5.895105552608323, −5.494173522749570, −4.631498732252588, −4.257077862325039, −3.724386635588498, −3.347774694225690, −2.470598175612578, −2.047162021857516, −1.425896061501242, −0.5498778755157254, 0,
0.5498778755157254, 1.425896061501242, 2.047162021857516, 2.470598175612578, 3.347774694225690, 3.724386635588498, 4.257077862325039, 4.631498732252588, 5.494173522749570, 5.895105552608323, 6.090189001947663, 6.618133333841738, 7.275620583337451, 7.655769416543019, 8.061238059320114, 8.730730078912845, 9.216069971700191, 9.613108014566733, 10.18748667622353, 10.29551819630084, 11.12963666617640, 11.30445637789705, 11.86866205673183, 12.25188516049543, 12.87734060775100
