| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 4·11-s + 2·13-s + 15-s + 2·17-s + 2·19-s + 21-s − 8·23-s + 25-s − 27-s + 4·29-s − 8·31-s − 4·33-s + 35-s + 6·37-s − 2·39-s − 6·41-s + 4·43-s − 45-s + 10·47-s + 49-s − 2·51-s − 53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.458·19-s + 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s − 1.43·31-s − 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 1.45·47-s + 1/7·49-s − 0.280·51-s − 0.137·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 53 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 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
−14.91557814065333, −14.30137538101363, −13.97572266649676, −13.40926625290428, −12.63932501753995, −12.20816018450683, −11.95857295060619, −11.25151013110989, −10.93304780340176, −10.17362478870252, −9.755617867115248, −9.144023657672502, −8.670401096707619, −7.892183075349825, −7.478915521789057, −6.809362803262329, −6.203950661618263, −5.894118758426025, −5.171830592879955, −4.365819917573434, −3.854379190170392, −3.468986138413737, −2.508396423644255, −1.580942428083487, −0.9538748137391237, 0,
0.9538748137391237, 1.580942428083487, 2.508396423644255, 3.468986138413737, 3.854379190170392, 4.365819917573434, 5.171830592879955, 5.894118758426025, 6.203950661618263, 6.809362803262329, 7.478915521789057, 7.892183075349825, 8.670401096707619, 9.144023657672502, 9.755617867115248, 10.17362478870252, 10.93304780340176, 11.25151013110989, 11.95857295060619, 12.20816018450683, 12.63932501753995, 13.40926625290428, 13.97572266649676, 14.30137538101363, 14.91557814065333
