| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s − 4·11-s − 4·13-s + 2·15-s + 4·17-s − 6·19-s − 21-s + 23-s − 25-s − 27-s − 2·29-s − 2·31-s + 4·33-s − 2·35-s + 2·37-s + 4·39-s − 10·41-s + 8·43-s − 2·45-s − 6·47-s + 49-s − 4·51-s + 6·53-s + 8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.516·15-s + 0.970·17-s − 1.37·19-s − 0.218·21-s + 0.208·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.359·31-s + 0.696·33-s − 0.338·35-s + 0.328·37-s + 0.640·39-s − 1.56·41-s + 1.21·43-s − 0.298·45-s − 0.875·47-s + 1/7·49-s − 0.560·51-s + 0.824·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−15.30699904492425, −14.98582330819206, −14.46105141077404, −13.79193442869856, −13.00920274167764, −12.64194039328246, −12.24581237815923, −11.54504103547636, −11.22673255485728, −10.57450436830859, −10.08549406229840, −9.648024308200667, −8.681460048848564, −8.168740393057530, −7.701958390876559, −7.235091128483510, −6.615114481530313, −5.780112898703784, −5.178738954639804, −4.836518413147834, −4.029295793584844, −3.491417676108231, −2.502873574176564, −1.974673395105142, −0.7291606916264190, 0,
0.7291606916264190, 1.974673395105142, 2.502873574176564, 3.491417676108231, 4.029295793584844, 4.836518413147834, 5.178738954639804, 5.780112898703784, 6.615114481530313, 7.235091128483510, 7.701958390876559, 8.168740393057530, 8.681460048848564, 9.648024308200667, 10.08549406229840, 10.57450436830859, 11.22673255485728, 11.54504103547636, 12.24581237815923, 12.64194039328246, 13.00920274167764, 13.79193442869856, 14.46105141077404, 14.98582330819206, 15.30699904492425
