| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 4·11-s − 15-s − 6·17-s − 4·19-s − 2·21-s − 4·23-s + 25-s − 27-s − 6·29-s − 10·31-s − 4·33-s + 2·35-s − 4·37-s − 10·41-s − 4·43-s + 45-s + 4·47-s − 3·49-s + 6·51-s + 10·53-s + 4·55-s + 4·57-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.436·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.79·31-s − 0.696·33-s + 0.338·35-s − 0.657·37-s − 1.56·41-s − 0.609·43-s + 0.149·45-s + 0.583·47-s − 3/7·49-s + 0.840·51-s + 1.37·53-s + 0.539·55-s + 0.529·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.270917125474520850097840052750, −7.09729515256919282004979746314, −6.72124935793995043616909418558, −5.85966818538757126585632437137, −5.18917018861302563909076214060, −4.28438537927342418401479730208, −3.71374910316435715607958883017, −2.08955780119710972105737730244, −1.62807832443618618694487017392, 0,
1.62807832443618618694487017392, 2.08955780119710972105737730244, 3.71374910316435715607958883017, 4.28438537927342418401479730208, 5.18917018861302563909076214060, 5.85966818538757126585632437137, 6.72124935793995043616909418558, 7.09729515256919282004979746314, 8.270917125474520850097840052750
