| L(s) = 1 | − 2·3-s + 2·5-s + 9-s + 11-s − 2·13-s − 4·15-s − 2·19-s − 25-s + 4·27-s − 6·29-s − 4·31-s − 2·33-s − 2·37-s + 4·39-s + 8·41-s + 12·43-s + 2·45-s − 12·47-s + 2·53-s + 2·55-s + 4·57-s + 10·59-s + 10·61-s − 4·65-s − 12·67-s − 4·71-s + 12·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 1.03·15-s − 0.458·19-s − 1/5·25-s + 0.769·27-s − 1.11·29-s − 0.718·31-s − 0.348·33-s − 0.328·37-s + 0.640·39-s + 1.24·41-s + 1.82·43-s + 0.298·45-s − 1.75·47-s + 0.274·53-s + 0.269·55-s + 0.529·57-s + 1.30·59-s + 1.28·61-s − 0.496·65-s − 1.46·67-s − 0.474·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.087809027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.087809027\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.77261632178363, −14.49289169244896, −13.99926447836632, −13.17071938086219, −12.88844846062249, −12.32558645192006, −11.76223115457718, −11.16269742538357, −10.89452360009280, −10.19017457925866, −9.679348414603465, −9.247302123742622, −8.581341305102920, −7.845276660318594, −7.109205938692827, −6.711875537337441, −5.877655777067657, −5.731316369775053, −5.156441771353557, −4.392759514388098, −3.813927809272442, −2.787653665144339, −2.133064720048665, −1.391846481351043, −0.4250305089556277,
0.4250305089556277, 1.391846481351043, 2.133064720048665, 2.787653665144339, 3.813927809272442, 4.392759514388098, 5.156441771353557, 5.731316369775053, 5.877655777067657, 6.711875537337441, 7.109205938692827, 7.845276660318594, 8.581341305102920, 9.247302123742622, 9.679348414603465, 10.19017457925866, 10.89452360009280, 11.16269742538357, 11.76223115457718, 12.32558645192006, 12.88844846062249, 13.17071938086219, 13.99926447836632, 14.49289169244896, 14.77261632178363
