| L(s) = 1 | + 3·3-s − 5-s + 6·9-s − 4·11-s − 6·13-s − 3·15-s − 3·17-s + 4·19-s + 2·23-s − 4·25-s + 9·27-s − 29-s − 9·31-s − 12·33-s + 8·37-s − 18·39-s + 41-s + 5·43-s − 6·45-s + 6·47-s − 9·51-s + 3·53-s + 4·55-s + 12·57-s + 14·59-s − 11·61-s + 6·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 2·9-s − 1.20·11-s − 1.66·13-s − 0.774·15-s − 0.727·17-s + 0.917·19-s + 0.417·23-s − 4/5·25-s + 1.73·27-s − 0.185·29-s − 1.61·31-s − 2.08·33-s + 1.31·37-s − 2.88·39-s + 0.156·41-s + 0.762·43-s − 0.894·45-s + 0.875·47-s − 1.26·51-s + 0.412·53-s + 0.539·55-s + 1.58·57-s + 1.82·59-s − 1.40·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−13.79225206271695, −13.29995761844917, −12.88435613395262, −12.46974504811034, −11.98737156827718, −11.19706451138706, −10.86449859927256, −10.12352493476369, −9.731397637798590, −9.278049673294072, −8.990124332129049, −8.216323811483688, −7.713046142421357, −7.605850441659873, −7.158915280560428, −6.497442233011631, −5.444383463253873, −5.200370685824034, −4.466549176029197, −3.909015311433219, −3.469716419939883, −2.648282235276347, −2.437666352238117, −1.982725957874190, −0.8876880690829983, 0,
0.8876880690829983, 1.982725957874190, 2.437666352238117, 2.648282235276347, 3.469716419939883, 3.909015311433219, 4.466549176029197, 5.200370685824034, 5.444383463253873, 6.497442233011631, 7.158915280560428, 7.605850441659873, 7.713046142421357, 8.216323811483688, 8.990124332129049, 9.278049673294072, 9.731397637798590, 10.12352493476369, 10.86449859927256, 11.19706451138706, 11.98737156827718, 12.46974504811034, 12.88435613395262, 13.29995761844917, 13.79225206271695
