| L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s + 2·13-s + 2·15-s − 17-s − 4·19-s + 4·21-s + 4·23-s − 25-s − 27-s + 6·29-s + 8·31-s + 8·35-s + 2·37-s − 2·39-s + 6·41-s − 4·43-s − 2·45-s − 4·47-s + 9·49-s + 51-s − 2·53-s + 4·57-s − 4·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s − 0.242·17-s − 0.917·19-s + 0.872·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 1.35·35-s + 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s − 0.583·47-s + 9/7·49-s + 0.140·51-s − 0.274·53-s + 0.529·57-s − 0.520·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 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.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−12.66923208376761, −12.29547239312563, −11.73273924385600, −11.32167696875784, −10.97554366276307, −10.42158363638799, −9.990551783413852, −9.611681303844562, −9.103450621072074, −8.484138228561785, −8.180327498385795, −7.683018773574245, −6.900062387208125, −6.671748218687275, −6.363593193740082, −5.867328773389864, −5.208410765898989, −4.616953850123732, −4.223414189800599, −3.661103002803599, −3.269562232370437, −2.664340745836424, −2.122594473996310, −1.065076250228320, −0.6625507237108609, 0,
0.6625507237108609, 1.065076250228320, 2.122594473996310, 2.664340745836424, 3.269562232370437, 3.661103002803599, 4.223414189800599, 4.616953850123732, 5.208410765898989, 5.867328773389864, 6.363593193740082, 6.671748218687275, 6.900062387208125, 7.683018773574245, 8.180327498385795, 8.484138228561785, 9.103450621072074, 9.611681303844562, 9.990551783413852, 10.42158363638799, 10.97554366276307, 11.32167696875784, 11.73273924385600, 12.29547239312563, 12.66923208376761
