| L(s) = 1 | − 3-s + 4·5-s + 9-s − 11-s + 6·13-s − 4·15-s − 2·19-s + 4·23-s + 11·25-s − 27-s + 2·29-s + 2·31-s + 33-s − 2·37-s − 6·39-s + 4·43-s + 4·45-s − 6·47-s − 2·53-s − 4·55-s + 2·57-s + 14·61-s + 24·65-s + 12·67-s − 4·69-s + 8·71-s + 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 1.03·15-s − 0.458·19-s + 0.834·23-s + 11/5·25-s − 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.174·33-s − 0.328·37-s − 0.960·39-s + 0.609·43-s + 0.596·45-s − 0.875·47-s − 0.274·53-s − 0.539·55-s + 0.264·57-s + 1.79·61-s + 2.97·65-s + 1.46·67-s − 0.481·69-s + 0.949·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.318505267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.318505267\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.80623655393184, −13.15428111522453, −12.79267689411532, −12.58390804289591, −11.52214155250464, −11.27587908966998, −10.70592761279774, −10.29403127617208, −9.919215336350270, −9.278152371830107, −8.858598204740116, −8.391049999347397, −7.750603846972913, −6.852995426597072, −6.501780396265665, −6.178158346733563, −5.609211970634460, −5.118458991522812, −4.710465357437873, −3.771307376654926, −3.282031907145412, −2.375943272994839, −1.999196933693110, −1.167347367704277, −0.7604005431409726,
0.7604005431409726, 1.167347367704277, 1.999196933693110, 2.375943272994839, 3.282031907145412, 3.771307376654926, 4.710465357437873, 5.118458991522812, 5.609211970634460, 6.178158346733563, 6.501780396265665, 6.852995426597072, 7.750603846972913, 8.391049999347397, 8.858598204740116, 9.278152371830107, 9.919215336350270, 10.29403127617208, 10.70592761279774, 11.27587908966998, 11.52214155250464, 12.58390804289591, 12.79267689411532, 13.15428111522453, 13.80623655393184
