| L(s) = 1 | + 2·3-s + 2·5-s + 9-s + 11-s + 4·15-s − 4·17-s + 4·19-s − 4·23-s − 25-s − 4·27-s − 2·29-s + 2·31-s + 2·33-s + 6·37-s − 4·41-s + 4·43-s + 2·45-s − 2·47-s − 8·51-s − 2·53-s + 2·55-s + 8·57-s − 6·59-s + 4·61-s − 8·69-s − 12·71-s − 16·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 1/3·9-s + 0.301·11-s + 1.03·15-s − 0.970·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s − 0.769·27-s − 0.371·29-s + 0.359·31-s + 0.348·33-s + 0.986·37-s − 0.624·41-s + 0.609·43-s + 0.298·45-s − 0.291·47-s − 1.12·51-s − 0.274·53-s + 0.269·55-s + 1.05·57-s − 0.781·59-s + 0.512·61-s − 0.963·69-s − 1.42·71-s − 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−15.21980573945876, −14.47115356156796, −14.16529745964268, −13.79071006985428, −13.15995099181747, −12.98937187728033, −12.03146242795867, −11.52768953385912, −11.05648229321733, −10.06625120910057, −9.907982346009284, −9.306525113343176, −8.771239327398753, −8.418684874375544, −7.556175271743257, −7.315476020298016, −6.309484234805174, −5.982052408409717, −5.311278663358708, −4.426632462459894, −3.955234034854684, −3.081085719512506, −2.636895041046299, −1.917015266413491, −1.364023431659138, 0,
1.364023431659138, 1.917015266413491, 2.636895041046299, 3.081085719512506, 3.955234034854684, 4.426632462459894, 5.311278663358708, 5.982052408409717, 6.309484234805174, 7.315476020298016, 7.556175271743257, 8.418684874375544, 8.771239327398753, 9.306525113343176, 9.907982346009284, 10.06625120910057, 11.05648229321733, 11.52768953385912, 12.03146242795867, 12.98937187728033, 13.15995099181747, 13.79071006985428, 14.16529745964268, 14.47115356156796, 15.21980573945876
