| L(s) = 1 | − 3-s + 9-s + 4·11-s + 4·13-s − 17-s − 19-s − 8·23-s − 27-s + 6·31-s − 4·33-s − 2·37-s − 4·39-s − 8·43-s + 8·47-s − 7·49-s + 51-s + 4·53-s + 57-s − 10·61-s + 2·67-s + 8·69-s + 10·71-s − 2·73-s − 6·79-s + 81-s − 4·83-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.242·17-s − 0.229·19-s − 1.66·23-s − 0.192·27-s + 1.07·31-s − 0.696·33-s − 0.328·37-s − 0.640·39-s − 1.21·43-s + 1.16·47-s − 49-s + 0.140·51-s + 0.549·53-s + 0.132·57-s − 1.28·61-s + 0.244·67-s + 0.963·69-s + 1.18·71-s − 0.234·73-s − 0.675·79-s + 1/9·81-s − 0.439·83-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−12.64132209751933, −12.02563021758643, −11.83551565689435, −11.41986954936888, −10.95137661094967, −10.42100564522605, −10.05686209232965, −9.588038671515792, −9.053884587052293, −8.553553113505548, −8.247653487989414, −7.664826215196506, −7.064060567876499, −6.502569967065144, −6.295300727731471, −5.865127504355037, −5.325627265129325, −4.552430945917656, −4.316504677663387, −3.685441469920260, −3.380903941262281, −2.509125220670108, −1.837395156335612, −1.404654600835541, −0.7730833008183248, 0,
0.7730833008183248, 1.404654600835541, 1.837395156335612, 2.509125220670108, 3.380903941262281, 3.685441469920260, 4.316504677663387, 4.552430945917656, 5.325627265129325, 5.865127504355037, 6.295300727731471, 6.502569967065144, 7.064060567876499, 7.664826215196506, 8.247653487989414, 8.553553113505548, 9.053884587052293, 9.588038671515792, 10.05686209232965, 10.42100564522605, 10.95137661094967, 11.41986954936888, 11.83551565689435, 12.02563021758643, 12.64132209751933
