| L(s) = 1 | − 3-s + 4·5-s + 9-s − 13-s − 4·15-s − 4·19-s − 4·23-s + 11·25-s − 27-s + 8·29-s − 8·31-s + 4·37-s + 39-s − 4·41-s − 4·43-s + 4·45-s − 8·47-s − 7·49-s + 10·53-s + 4·57-s + 4·59-s − 8·61-s − 4·65-s + 4·67-s + 4·69-s + 4·73-s − 11·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1/3·9-s − 0.277·13-s − 1.03·15-s − 0.917·19-s − 0.834·23-s + 11/5·25-s − 0.192·27-s + 1.48·29-s − 1.43·31-s + 0.657·37-s + 0.160·39-s − 0.624·41-s − 0.609·43-s + 0.596·45-s − 1.16·47-s − 49-s + 1.37·53-s + 0.529·57-s + 0.520·59-s − 1.02·61-s − 0.496·65-s + 0.488·67-s + 0.481·69-s + 0.468·73-s − 1.27·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.351080619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.351080619\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−13.14061846741757, −12.81460108561313, −12.18789346139164, −11.91230853328901, −11.09530957969219, −10.74369061122415, −10.31154645285915, −9.849014685975095, −9.518793802032787, −9.048391459760046, −8.278277622220743, −8.104494280904162, −7.085184556063390, −6.677102537372564, −6.411500008501786, −5.723619281125673, −5.504337605992260, −4.834735896982708, −4.458127984805305, −3.652484706653076, −2.949581378529930, −2.286239403489488, −1.861487440569847, −1.325199890470012, −0.4416496316687089,
0.4416496316687089, 1.325199890470012, 1.861487440569847, 2.286239403489488, 2.949581378529930, 3.652484706653076, 4.458127984805305, 4.834735896982708, 5.504337605992260, 5.723619281125673, 6.411500008501786, 6.677102537372564, 7.085184556063390, 8.104494280904162, 8.278277622220743, 9.048391459760046, 9.518793802032787, 9.849014685975095, 10.31154645285915, 10.74369061122415, 11.09530957969219, 11.91230853328901, 12.18789346139164, 12.81460108561313, 13.14061846741757
