| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 11-s + 16-s + 17-s + 2·19-s + 20-s − 22-s + 3·23-s + 25-s + 5·29-s + 3·31-s − 32-s − 34-s + 37-s − 2·38-s − 40-s + 10·41-s − 11·43-s + 44-s − 3·46-s + 7·47-s − 7·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 1/4·16-s + 0.242·17-s + 0.458·19-s + 0.223·20-s − 0.213·22-s + 0.625·23-s + 1/5·25-s + 0.928·29-s + 0.538·31-s − 0.176·32-s − 0.171·34-s + 0.164·37-s − 0.324·38-s − 0.158·40-s + 1.56·41-s − 1.67·43-s + 0.150·44-s − 0.442·46-s + 1.02·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−12.89538736494913, −12.62793384257031, −12.02097156302344, −11.63533121126924, −11.10517053109446, −10.77442006888172, −10.04214774435733, −9.899946649106623, −9.408654462316912, −8.782949553795504, −8.568650487964724, −7.911077782184030, −7.460936383247164, −6.981096953295441, −6.432098471395562, −6.069293745283798, −5.509732176018792, −4.867271910646833, −4.490860284369975, −3.654577938045877, −3.150031157293080, −2.604322594673483, −2.055879924806835, −1.228828327718385, −0.9699455483985473, 0,
0.9699455483985473, 1.228828327718385, 2.055879924806835, 2.604322594673483, 3.150031157293080, 3.654577938045877, 4.490860284369975, 4.867271910646833, 5.509732176018792, 6.069293745283798, 6.432098471395562, 6.981096953295441, 7.460936383247164, 7.911077782184030, 8.568650487964724, 8.782949553795504, 9.408654462316912, 9.899946649106623, 10.04214774435733, 10.77442006888172, 11.10517053109446, 11.63533121126924, 12.02097156302344, 12.62793384257031, 12.89538736494913
