| L(s) = 1 | + 3-s − 2·5-s + 9-s + 11-s + 2·13-s − 2·15-s − 2·17-s + 4·19-s − 25-s + 27-s − 6·29-s + 4·31-s + 33-s − 2·37-s + 2·39-s + 6·41-s + 4·43-s − 2·45-s − 8·47-s − 2·51-s − 6·53-s − 2·55-s + 4·57-s + 12·59-s + 10·61-s − 4·65-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.174·33-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s − 0.280·51-s − 0.824·53-s − 0.269·55-s + 0.529·57-s + 1.56·59-s + 1.28·61-s − 0.496·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.74363840057468, −14.28198810021630, −13.73595308797507, −13.20188468672435, −12.80469367531662, −12.14404861030968, −11.52222758077270, −11.34190955690750, −10.71227408918229, −9.898519971275222, −9.615576146666879, −8.848248113201563, −8.544845640735447, −7.837322448920371, −7.526090162314431, −6.906068187986446, −6.314554067471434, −5.605440137336486, −4.982898136149467, −4.093276717762051, −3.949598573412754, −3.200843879071279, −2.597727500565553, −1.725725943805248, −1.001769286249007, 0,
1.001769286249007, 1.725725943805248, 2.597727500565553, 3.200843879071279, 3.949598573412754, 4.093276717762051, 4.982898136149467, 5.605440137336486, 6.314554067471434, 6.906068187986446, 7.526090162314431, 7.837322448920371, 8.544845640735447, 8.848248113201563, 9.615576146666879, 9.898519971275222, 10.71227408918229, 11.34190955690750, 11.52222758077270, 12.14404861030968, 12.80469367531662, 13.20188468672435, 13.73595308797507, 14.28198810021630, 14.74363840057468
