| L(s) = 1 | − 3-s + 2·5-s − 2·7-s + 9-s − 11-s − 2·15-s − 17-s − 6·19-s + 2·21-s − 6·23-s − 25-s − 27-s + 2·29-s + 4·31-s + 33-s − 4·35-s + 10·37-s − 6·41-s + 10·43-s + 2·45-s − 12·47-s − 3·49-s + 51-s + 4·53-s − 2·55-s + 6·57-s + 10·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.516·15-s − 0.242·17-s − 1.37·19-s + 0.436·21-s − 1.25·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.174·33-s − 0.676·35-s + 1.64·37-s − 0.937·41-s + 1.52·43-s + 0.298·45-s − 1.75·47-s − 3/7·49-s + 0.140·51-s + 0.549·53-s − 0.269·55-s + 0.794·57-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.36661766458358, −14.54710835892318, −14.13385958687559, −13.41710264450499, −13.13023280953516, −12.63256769542397, −12.10890928787475, −11.46137030229543, −10.90562402784624, −10.33086930829039, −9.832439874128912, −9.598792463416252, −8.801579608303390, −8.138612851758866, −7.671824630126908, −6.646828901619362, −6.381936779205136, −6.026689097328423, −5.317644235919447, −4.626557404068776, −4.053129819820539, −3.283500940710496, −2.312040539873648, −2.042928071440942, −0.8833430448813024, 0,
0.8833430448813024, 2.042928071440942, 2.312040539873648, 3.283500940710496, 4.053129819820539, 4.626557404068776, 5.317644235919447, 6.026689097328423, 6.381936779205136, 6.646828901619362, 7.671824630126908, 8.138612851758866, 8.801579608303390, 9.598792463416252, 9.832439874128912, 10.33086930829039, 10.90562402784624, 11.46137030229543, 12.10890928787475, 12.63256769542397, 13.13023280953516, 13.41710264450499, 14.13385958687559, 14.54710835892318, 15.36661766458358
