| L(s) = 1 | + 3-s + 2·5-s − 4·7-s + 9-s + 2·15-s − 2·17-s + 6·19-s − 4·21-s − 8·23-s − 25-s + 27-s − 4·29-s − 4·31-s − 8·35-s + 6·37-s − 8·43-s + 2·45-s + 9·49-s − 2·51-s + 6·57-s + 8·59-s + 10·61-s − 4·63-s + 14·67-s − 8·69-s + 8·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.516·15-s − 0.485·17-s + 1.37·19-s − 0.872·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 0.742·29-s − 0.718·31-s − 1.35·35-s + 0.986·37-s − 1.21·43-s + 0.298·45-s + 9/7·49-s − 0.280·51-s + 0.794·57-s + 1.04·59-s + 1.28·61-s − 0.503·63-s + 1.71·67-s − 0.963·69-s + 0.949·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.52958038657213, −13.73472475317353, −13.46400471253388, −13.24516720702782, −12.50547665746361, −12.16841863374001, −11.42676944692504, −10.91316906807466, −10.00419971305839, −9.836061304374459, −9.547045055464924, −9.067343606837415, −8.258907846575872, −7.858855352357485, −7.083484924750797, −6.658133262803859, −6.145932572829141, −5.548853034748993, −5.138941675930295, −4.004714116328390, −3.717495186629058, −3.098134683412610, −2.298624245255489, −1.997520719267049, −0.9461887930831782, 0,
0.9461887930831782, 1.997520719267049, 2.298624245255489, 3.098134683412610, 3.717495186629058, 4.004714116328390, 5.138941675930295, 5.548853034748993, 6.145932572829141, 6.658133262803859, 7.083484924750797, 7.858855352357485, 8.258907846575872, 9.067343606837415, 9.547045055464924, 9.836061304374459, 10.00419971305839, 10.91316906807466, 11.42676944692504, 12.16841863374001, 12.50547665746361, 13.24516720702782, 13.46400471253388, 13.73472475317353, 14.52958038657213
