| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 11-s + 15-s − 5·17-s + 2·19-s − 21-s + 8·23-s + 25-s − 27-s + 5·29-s + 2·31-s + 33-s − 35-s − 10·41-s + 4·43-s − 45-s + 3·47-s + 49-s + 5·51-s − 6·53-s + 55-s − 2·57-s + 4·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.258·15-s − 1.21·17-s + 0.458·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.928·29-s + 0.359·31-s + 0.174·33-s − 0.169·35-s − 1.56·41-s + 0.609·43-s − 0.149·45-s + 0.437·47-s + 1/7·49-s + 0.700·51-s − 0.824·53-s + 0.134·55-s − 0.264·57-s + 0.512·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−14.45207983076721, −13.75835964745799, −13.31094353696263, −12.92653162862346, −12.28058132032855, −11.76199602035846, −11.46776431834597, −10.76223698953707, −10.61582963315136, −9.930404407598441, −9.165546017867688, −8.851138656029905, −8.200466939985681, −7.703181109469875, −6.965574262749081, −6.782725203950910, −6.055880129044964, −5.365543162713059, −4.782733677939379, −4.564622657572243, −3.733093723401489, −3.026370393380639, −2.450766211945086, −1.533769124728409, −0.8627580449838067, 0,
0.8627580449838067, 1.533769124728409, 2.450766211945086, 3.026370393380639, 3.733093723401489, 4.564622657572243, 4.782733677939379, 5.365543162713059, 6.055880129044964, 6.782725203950910, 6.965574262749081, 7.703181109469875, 8.200466939985681, 8.851138656029905, 9.165546017867688, 9.930404407598441, 10.61582963315136, 10.76223698953707, 11.46776431834597, 11.76199602035846, 12.28058132032855, 12.92653162862346, 13.31094353696263, 13.75835964745799, 14.45207983076721
