| L(s) = 1 | − 2·2-s + 2·4-s + 3·5-s − 7-s − 3·9-s − 6·10-s + 3·11-s + 2·14-s − 4·16-s − 3·17-s + 6·18-s + 19-s + 6·20-s − 6·22-s + 8·23-s + 4·25-s − 2·28-s + 2·29-s + 2·31-s + 8·32-s + 6·34-s − 3·35-s − 6·36-s − 4·37-s − 2·38-s − 6·41-s − 5·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.34·5-s − 0.377·7-s − 9-s − 1.89·10-s + 0.904·11-s + 0.534·14-s − 16-s − 0.727·17-s + 1.41·18-s + 0.229·19-s + 1.34·20-s − 1.27·22-s + 1.66·23-s + 4/5·25-s − 0.377·28-s + 0.371·29-s + 0.359·31-s + 1.41·32-s + 1.02·34-s − 0.507·35-s − 36-s − 0.657·37-s − 0.324·38-s − 0.937·41-s − 0.762·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53371 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9978180841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9978180841\) |
| \(L(\frac{3}{2})\) |
|
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 | 19 | \( 1 - T \) | |
| 53 | \( 1 \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.53909203080144, −13.67801488418822, −13.59980559979639, −13.15172826341590, −12.11188822478844, −11.87961658931339, −11.10404657312045, −10.60116971449683, −10.36466795696918, −9.503863866549779, −9.265298668849551, −8.908018317037550, −8.486161044776413, −7.735403526473247, −6.994637255208036, −6.611228857690926, −6.156156094424114, −5.458574407481911, −4.873447227447098, −4.127360804479137, −2.967189061509616, −2.790276303433025, −1.723263948561855, −1.421605153386742, −0.4448877339189154,
0.4448877339189154, 1.421605153386742, 1.723263948561855, 2.790276303433025, 2.967189061509616, 4.127360804479137, 4.873447227447098, 5.458574407481911, 6.156156094424114, 6.611228857690926, 6.994637255208036, 7.735403526473247, 8.486161044776413, 8.908018317037550, 9.265298668849551, 9.503863866549779, 10.36466795696918, 10.60116971449683, 11.10404657312045, 11.87961658931339, 12.11188822478844, 13.15172826341590, 13.59980559979639, 13.67801488418822, 14.53909203080144
