| L(s) = 1 | + 2·3-s − 4·5-s − 4·7-s + 9-s − 2·11-s − 2·13-s − 8·15-s + 4·19-s − 8·21-s + 4·23-s + 11·25-s − 4·27-s − 2·29-s − 4·31-s − 4·33-s + 16·35-s + 2·37-s − 4·39-s − 4·41-s − 4·43-s − 4·45-s + 8·47-s + 9·49-s − 10·53-s + 8·55-s + 8·57-s − 12·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 2.06·15-s + 0.917·19-s − 1.74·21-s + 0.834·23-s + 11/5·25-s − 0.769·27-s − 0.371·29-s − 0.718·31-s − 0.696·33-s + 2.70·35-s + 0.328·37-s − 0.640·39-s − 0.624·41-s − 0.609·43-s − 0.596·45-s + 1.16·47-s + 9/7·49-s − 1.37·53-s + 1.07·55-s + 1.05·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8937847190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8937847190\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| 83 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−12.56290486411554, −12.08303596672845, −11.63436308664939, −11.10349897480911, −10.76515759765027, −10.03034035121655, −9.713146631189196, −9.152692805874992, −8.906602054745743, −8.327877623596181, −7.850063131056281, −7.471202398302364, −7.226844121151965, −6.665827723224463, −6.098939719494675, −5.324086026609846, −4.939725233876331, −4.266714867613377, −3.679690807424800, −3.362535626961727, −3.073097874409672, −2.603537028710119, −1.901840208481210, −0.8606680582008911, −0.2780096521275995,
0.2780096521275995, 0.8606680582008911, 1.901840208481210, 2.603537028710119, 3.073097874409672, 3.362535626961727, 3.679690807424800, 4.266714867613377, 4.939725233876331, 5.324086026609846, 6.098939719494675, 6.665827723224463, 7.226844121151965, 7.471202398302364, 7.850063131056281, 8.327877623596181, 8.906602054745743, 9.152692805874992, 9.713146631189196, 10.03034035121655, 10.76515759765027, 11.10349897480911, 11.63436308664939, 12.08303596672845, 12.56290486411554
