| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 12-s + 4·13-s − 14-s + 16-s + 17-s + 18-s + 2·19-s + 21-s − 6·23-s − 24-s + 4·26-s − 27-s − 28-s − 4·31-s + 32-s + 34-s + 36-s − 8·37-s + 2·38-s − 4·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.458·19-s + 0.218·21-s − 1.25·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.188·28-s − 0.718·31-s + 0.176·32-s + 0.171·34-s + 1/6·36-s − 1.31·37-s + 0.324·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−15.97526032809101, −15.86479178151990, −14.92356765419620, −14.38009302921189, −13.88543973655064, −13.15799016317257, −12.97745618315341, −12.13144072335015, −11.75466319447837, −11.24588097491250, −10.54946377376266, −10.10906287437735, −9.482130197748430, −8.623189251550939, −8.109696804618831, −7.283970942342169, −6.721218525209304, −6.185863465017439, −5.510372443029885, −5.166832361315347, −4.089703751710956, −3.748881725850038, −2.998935339508389, −1.969061874477017, −1.229760269697488, 0,
1.229760269697488, 1.969061874477017, 2.998935339508389, 3.748881725850038, 4.089703751710956, 5.166832361315347, 5.510372443029885, 6.185863465017439, 6.721218525209304, 7.283970942342169, 8.109696804618831, 8.623189251550939, 9.482130197748430, 10.10906287437735, 10.54946377376266, 11.24588097491250, 11.75466319447837, 12.13144072335015, 12.97745618315341, 13.15799016317257, 13.88543973655064, 14.38009302921189, 14.92356765419620, 15.86479178151990, 15.97526032809101
