| L(s) = 1 | − 2·5-s − 2·13-s − 2·17-s − 25-s − 4·31-s − 2·37-s − 6·41-s + 4·43-s − 4·47-s − 7·49-s − 6·53-s + 2·59-s + 2·61-s + 4·65-s − 4·67-s + 2·71-s − 10·73-s − 4·83-s + 4·85-s − 6·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.554·13-s − 0.485·17-s − 1/5·25-s − 0.718·31-s − 0.328·37-s − 0.937·41-s + 0.609·43-s − 0.583·47-s − 49-s − 0.824·53-s + 0.260·59-s + 0.256·61-s + 0.496·65-s − 0.488·67-s + 0.237·71-s − 1.17·73-s − 0.439·83-s + 0.433·85-s − 0.635·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.08493983369285, −12.67283151848673, −12.23589630553444, −11.69579495496300, −11.38590341962723, −10.98389535023353, −10.35146980934744, −9.943543602968130, −9.433164437599805, −8.879150776035968, −8.458250842389729, −7.943410084115801, −7.481245039367736, −7.102919032962766, −6.551532074686253, −6.006387612452606, −5.411872457220318, −4.818320922349323, −4.445977705196509, −3.850454102294557, −3.319196551473860, −2.855719030128218, −2.016237276542100, −1.603351781036215, −0.5826673434748358, 0,
0.5826673434748358, 1.603351781036215, 2.016237276542100, 2.855719030128218, 3.319196551473860, 3.850454102294557, 4.445977705196509, 4.818320922349323, 5.411872457220318, 6.006387612452606, 6.551532074686253, 7.102919032962766, 7.481245039367736, 7.943410084115801, 8.458250842389729, 8.879150776035968, 9.433164437599805, 9.943543602968130, 10.35146980934744, 10.98389535023353, 11.38590341962723, 11.69579495496300, 12.23589630553444, 12.67283151848673, 13.08493983369285
