| L(s) = 1 | − 2-s − 4-s + 4·5-s + 3·8-s − 3·9-s − 4·10-s + 4·11-s + 4·13-s − 16-s + 3·18-s − 6·19-s − 4·20-s − 4·22-s + 4·23-s + 11·25-s − 4·26-s + 6·29-s + 2·31-s − 5·32-s + 3·36-s + 6·38-s + 12·40-s + 6·41-s + 4·43-s − 4·44-s − 12·45-s − 4·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.78·5-s + 1.06·8-s − 9-s − 1.26·10-s + 1.20·11-s + 1.10·13-s − 1/4·16-s + 0.707·18-s − 1.37·19-s − 0.894·20-s − 0.852·22-s + 0.834·23-s + 11/5·25-s − 0.784·26-s + 1.11·29-s + 0.359·31-s − 0.883·32-s + 1/2·36-s + 0.973·38-s + 1.89·40-s + 0.937·41-s + 0.609·43-s − 0.603·44-s − 1.78·45-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67081 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67081 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.762454630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.762454630\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−14.01431280743589, −13.72358010430540, −13.43296011354469, −12.74576859420617, −12.29385455745119, −11.54073421978106, −10.83184932054189, −10.56891544106432, −10.19750021976585, −9.252471356434685, −9.131926372824879, −8.821466386739530, −8.345263913922423, −7.563108276417201, −6.728678811931123, −6.317472407125600, −5.890163024824277, −5.411393445523097, −4.551684309671031, −4.182054548771638, −3.259491610953185, −2.549060785284073, −1.912672052882517, −1.177741103832617, −0.7292773829235355,
0.7292773829235355, 1.177741103832617, 1.912672052882517, 2.549060785284073, 3.259491610953185, 4.182054548771638, 4.551684309671031, 5.411393445523097, 5.890163024824277, 6.317472407125600, 6.728678811931123, 7.563108276417201, 8.345263913922423, 8.821466386739530, 9.131926372824879, 9.252471356434685, 10.19750021976585, 10.56891544106432, 10.83184932054189, 11.54073421978106, 12.29385455745119, 12.74576859420617, 13.43296011354469, 13.72358010430540, 14.01431280743589
