| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 4·11-s + 13-s + 15-s + 2·17-s + 4·19-s + 21-s + 25-s + 27-s − 2·29-s − 8·31-s − 4·33-s + 35-s + 6·37-s + 39-s − 6·41-s − 4·43-s + 45-s + 49-s + 2·51-s + 6·53-s − 4·55-s + 4·57-s + 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.169·35-s + 0.986·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.141784427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.141784427\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 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.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.44093530860006, −14.94943981131804, −14.49762490915982, −13.89389898546801, −13.34027980518008, −13.05534236043038, −12.39206386579119, −11.68047218659729, −11.13476591514988, −10.49605116607448, −9.993546240977160, −9.495485973568511, −8.816602903183267, −8.276315703678912, −7.651025386189811, −7.282512913237539, −6.481513135108334, −5.505527357854855, −5.385934580296882, −4.544491834983459, −3.648214814374861, −3.114761887270748, −2.305986544102028, −1.700443882763373, −0.6953227592868731,
0.6953227592868731, 1.700443882763373, 2.305986544102028, 3.114761887270748, 3.648214814374861, 4.544491834983459, 5.385934580296882, 5.505527357854855, 6.481513135108334, 7.282512913237539, 7.651025386189811, 8.276315703678912, 8.816602903183267, 9.495485973568511, 9.993546240977160, 10.49605116607448, 11.13476591514988, 11.68047218659729, 12.39206386579119, 13.05534236043038, 13.34027980518008, 13.89389898546801, 14.49762490915982, 14.94943981131804, 15.44093530860006
