| L(s) = 1 | + 4·5-s − 7-s − 3·11-s + 5·13-s − 17-s + 6·19-s − 5·23-s + 11·25-s + 7·29-s + 8·31-s − 4·35-s − 2·37-s + 6·41-s + 11·43-s − 6·49-s − 10·53-s − 12·55-s + 20·65-s − 2·67-s − 2·71-s − 9·73-s + 3·77-s − 79-s + 11·83-s − 4·85-s − 8·89-s − 5·91-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.377·7-s − 0.904·11-s + 1.38·13-s − 0.242·17-s + 1.37·19-s − 1.04·23-s + 11/5·25-s + 1.29·29-s + 1.43·31-s − 0.676·35-s − 0.328·37-s + 0.937·41-s + 1.67·43-s − 6/7·49-s − 1.37·53-s − 1.61·55-s + 2.48·65-s − 0.244·67-s − 0.237·71-s − 1.05·73-s + 0.341·77-s − 0.112·79-s + 1.20·83-s − 0.433·85-s − 0.847·89-s − 0.524·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.372190254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.372190254\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−16.13792426647031, −16.03554220196910, −15.60614834603428, −14.46729669560424, −13.95208737177392, −13.72712685972493, −13.14399247888860, −12.65159246340873, −11.89994574560174, −11.12343507824501, −10.39958490613207, −10.16730125305227, −9.442491133647078, −9.011601360004105, −8.200130980421928, −7.621313436683202, −6.503084435887328, −6.259627249033199, −5.648316807526751, −5.033839506159921, −4.181346856272349, −3.032252412983730, −2.656170767551901, −1.649908566810519, −0.8960359909749903,
0.8960359909749903, 1.649908566810519, 2.656170767551901, 3.032252412983730, 4.181346856272349, 5.033839506159921, 5.648316807526751, 6.259627249033199, 6.503084435887328, 7.621313436683202, 8.200130980421928, 9.011601360004105, 9.442491133647078, 10.16730125305227, 10.39958490613207, 11.12343507824501, 11.89994574560174, 12.65159246340873, 13.14399247888860, 13.72712685972493, 13.95208737177392, 14.46729669560424, 15.60614834603428, 16.03554220196910, 16.13792426647031
