| L(s) = 1 | − 3-s − 4·5-s − 7-s + 9-s + 4·15-s + 2·17-s + 21-s − 4·23-s + 11·25-s − 27-s − 2·29-s + 8·31-s + 4·35-s − 8·37-s − 4·41-s − 8·43-s − 4·45-s + 49-s − 2·51-s − 6·53-s + 8·59-s + 10·61-s − 63-s − 8·67-s + 4·69-s − 4·73-s − 11·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s + 1.03·15-s + 0.485·17-s + 0.218·21-s − 0.834·23-s + 11/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.676·35-s − 1.31·37-s − 0.624·41-s − 1.21·43-s − 0.596·45-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 1.04·59-s + 1.28·61-s − 0.125·63-s − 0.977·67-s + 0.481·69-s − 0.468·73-s − 1.27·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5054367191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5054367191\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−12.88185259274809, −12.17090243698454, −11.93399242533160, −11.81168822440668, −11.14831770171989, −10.72207366759545, −10.13917746005127, −9.899648803439368, −9.144557266181578, −8.476612896739281, −8.267402142080957, −7.707746759315606, −7.287481401352139, −6.668299408454313, −6.472803591120260, −5.655116504549613, −5.122206449754423, −4.649262665734429, −4.122752559560449, −3.529711699570235, −3.330752388623245, −2.515002302864409, −1.675752781253507, −0.9089431228943927, −0.2584529052470594,
0.2584529052470594, 0.9089431228943927, 1.675752781253507, 2.515002302864409, 3.330752388623245, 3.529711699570235, 4.122752559560449, 4.649262665734429, 5.122206449754423, 5.655116504549613, 6.472803591120260, 6.668299408454313, 7.287481401352139, 7.707746759315606, 8.267402142080957, 8.476612896739281, 9.144557266181578, 9.899648803439368, 10.13917746005127, 10.72207366759545, 11.14831770171989, 11.81168822440668, 11.93399242533160, 12.17090243698454, 12.88185259274809
