| L(s) = 1 | − 3-s − 3·5-s − 7-s + 9-s + 3·15-s + 7·17-s − 6·19-s + 21-s − 4·23-s + 4·25-s − 27-s − 6·29-s − 2·31-s + 3·35-s + 2·37-s + 2·41-s − 5·43-s − 3·45-s − 3·47-s + 49-s − 7·51-s + 4·53-s + 6·57-s − 9·59-s − 4·61-s − 63-s + 13·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.774·15-s + 1.69·17-s − 1.37·19-s + 0.218·21-s − 0.834·23-s + 4/5·25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.507·35-s + 0.328·37-s + 0.312·41-s − 0.762·43-s − 0.447·45-s − 0.437·47-s + 1/7·49-s − 0.980·51-s + 0.549·53-s + 0.794·57-s − 1.17·59-s − 0.512·61-s − 0.125·63-s + 1.58·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3085832569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3085832569\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 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
−13.05766339634978, −12.54494201780684, −12.39027175410901, −11.86417632757414, −11.38300142852450, −10.99041512246292, −10.50499962239123, −9.945186562115140, −9.571347667102353, −8.896922593156930, −8.284913502419220, −7.865336987626644, −7.549224143440107, −6.967121473866043, −6.346170770135957, −5.971590318309965, −5.310819946425330, −4.842647103178308, −4.082728826753011, −3.778595343325726, −3.367239443451589, −2.533590478838260, −1.789719727291775, −1.020543168695251, −0.1941845076474393,
0.1941845076474393, 1.020543168695251, 1.789719727291775, 2.533590478838260, 3.367239443451589, 3.778595343325726, 4.082728826753011, 4.842647103178308, 5.310819946425330, 5.971590318309965, 6.346170770135957, 6.967121473866043, 7.549224143440107, 7.865336987626644, 8.284913502419220, 8.896922593156930, 9.571347667102353, 9.945186562115140, 10.50499962239123, 10.99041512246292, 11.38300142852450, 11.86417632757414, 12.39027175410901, 12.54494201780684, 13.05766339634978
