| L(s) = 1 | − 3-s − 5-s + 9-s + 4·11-s − 2·13-s + 15-s − 4·19-s + 25-s − 27-s + 2·29-s + 8·31-s − 4·33-s − 6·37-s + 2·39-s + 6·41-s + 4·43-s − 45-s − 7·49-s − 10·53-s − 4·55-s + 4·57-s + 4·59-s + 2·61-s + 2·65-s − 4·67-s + 6·73-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.258·15-s − 0.917·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 49-s − 1.37·53-s − 0.539·55-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.248·65-s − 0.488·67-s + 0.702·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.525387734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.525387734\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.31161644251691, −13.67312696728252, −13.03864873304250, −12.47577196720784, −12.11804942640518, −11.75401786302816, −11.11497925384404, −10.76347626817846, −10.13187574816991, −9.589117543280083, −9.133215195451254, −8.468958214646603, −8.021789854696372, −7.365807805980195, −6.788986782714949, −6.343259536399018, −5.971855155768062, −4.969317283433908, −4.709256371027651, −4.046015761926418, −3.534833738438302, −2.722560285593829, −1.981039029331191, −1.193872338673179, −0.4686591035519483,
0.4686591035519483, 1.193872338673179, 1.981039029331191, 2.722560285593829, 3.534833738438302, 4.046015761926418, 4.709256371027651, 4.969317283433908, 5.971855155768062, 6.343259536399018, 6.788986782714949, 7.365807805980195, 8.021789854696372, 8.468958214646603, 9.133215195451254, 9.589117543280083, 10.13187574816991, 10.76347626817846, 11.11497925384404, 11.75401786302816, 12.11804942640518, 12.47577196720784, 13.03864873304250, 13.67312696728252, 14.31161644251691
