| L(s) = 1 | − 3-s − 2·5-s + 9-s + 11-s + 2·13-s + 2·15-s + 6·17-s + 4·19-s + 4·23-s − 25-s − 27-s + 6·29-s − 8·31-s − 33-s − 37-s − 2·39-s + 2·41-s + 4·43-s − 2·45-s − 7·49-s − 6·51-s − 6·53-s − 2·55-s − 4·57-s + 8·59-s + 10·61-s − 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.516·15-s + 1.45·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.174·33-s − 0.164·37-s − 0.320·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s − 49-s − 0.840·51-s − 0.824·53-s − 0.269·55-s − 0.529·57-s + 1.04·59-s + 1.28·61-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.36522029261415, −13.81729476956507, −13.14748352025858, −12.67914359509730, −12.17063475750548, −11.81533776807307, −11.28967540597383, −10.93770183794146, −10.35957504907249, −9.669408957586779, −9.405600465135456, −8.587281429451231, −8.120395709165031, −7.575508389718153, −7.179179863566769, −6.559689906149565, −5.976697447842227, −5.287119879350307, −5.049675635355796, −4.152905747904451, −3.638764566984400, −3.274248317994365, −2.409367850846878, −1.303629784673920, −0.9880632398007781, 0,
0.9880632398007781, 1.303629784673920, 2.409367850846878, 3.274248317994365, 3.638764566984400, 4.152905747904451, 5.049675635355796, 5.287119879350307, 5.976697447842227, 6.559689906149565, 7.179179863566769, 7.575508389718153, 8.120395709165031, 8.587281429451231, 9.405600465135456, 9.669408957586779, 10.35957504907249, 10.93770183794146, 11.28967540597383, 11.81533776807307, 12.17063475750548, 12.67914359509730, 13.14748352025858, 13.81729476956507, 14.36522029261415
