| L(s) = 1 | + 3-s − 2·5-s + 9-s − 13-s − 2·15-s − 6·17-s − 4·23-s − 25-s + 27-s − 6·29-s − 4·31-s + 2·37-s − 39-s − 10·41-s − 4·43-s − 2·45-s − 8·47-s − 7·49-s − 6·51-s − 6·53-s − 4·59-s + 2·61-s + 2·65-s + 4·67-s − 4·69-s + 8·71-s − 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.277·13-s − 0.516·15-s − 1.45·17-s − 0.834·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.328·37-s − 0.160·39-s − 1.56·41-s − 0.609·43-s − 0.298·45-s − 1.16·47-s − 49-s − 0.840·51-s − 0.824·53-s − 0.520·59-s + 0.256·61-s + 0.248·65-s + 0.488·67-s − 0.481·69-s + 0.949·71-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 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
−14.60890396477115, −14.11242386045217, −13.49822022617559, −13.06961255217638, −12.70027851994125, −11.99758021832065, −11.49242184213988, −11.22552637016903, −10.58500855095626, −9.911040446982773, −9.513953542965430, −8.946122519943422, −8.336090939657879, −8.034795221691869, −7.480090573676147, −6.884357437753764, −6.484297164966227, −5.720996211764200, −4.977582028768819, −4.491035881302174, −3.872456142376309, −3.471036401621559, −2.762083288028685, −1.963628645575468, −1.547704181728818, 0, 0,
1.547704181728818, 1.963628645575468, 2.762083288028685, 3.471036401621559, 3.872456142376309, 4.491035881302174, 4.977582028768819, 5.720996211764200, 6.484297164966227, 6.884357437753764, 7.480090573676147, 8.034795221691869, 8.336090939657879, 8.946122519943422, 9.513953542965430, 9.911040446982773, 10.58500855095626, 11.22552637016903, 11.49242184213988, 11.99758021832065, 12.70027851994125, 13.06961255217638, 13.49822022617559, 14.11242386045217, 14.60890396477115
