L(s) = 1 | + (−0.835 + 0.549i)2-s + (0.973 − 0.230i)3-s + (0.396 − 0.918i)4-s + (−0.686 + 0.727i)6-s + (0.173 + 0.984i)8-s + (0.893 − 0.448i)9-s + (−0.227 + 0.758i)11-s + (0.173 − 0.984i)12-s + (−0.686 − 0.727i)16-s + (0.109 − 0.0397i)17-s + (−0.499 + 0.866i)18-s + (0.539 + 0.196i)19-s + (−0.227 − 0.758i)22-s + (0.396 + 0.918i)24-s + (−0.0581 − 0.998i)25-s + ⋯ |
L(s) = 1 | + (−0.835 + 0.549i)2-s + (0.973 − 0.230i)3-s + (0.396 − 0.918i)4-s + (−0.686 + 0.727i)6-s + (0.173 + 0.984i)8-s + (0.893 − 0.448i)9-s + (−0.227 + 0.758i)11-s + (0.173 − 0.984i)12-s + (−0.686 − 0.727i)16-s + (0.109 − 0.0397i)17-s + (−0.499 + 0.866i)18-s + (0.539 + 0.196i)19-s + (−0.227 − 0.758i)22-s + (0.396 + 0.918i)24-s + (−0.0581 − 0.998i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8840058431\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8840058431\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.835 - 0.549i)T \) |
| 3 | \( 1 + (-0.973 + 0.230i)T \) |
good | 5 | \( 1 + (0.0581 + 0.998i)T^{2} \) |
| 7 | \( 1 + (-0.973 - 0.230i)T^{2} \) |
| 11 | \( 1 + (0.227 - 0.758i)T + (-0.835 - 0.549i)T^{2} \) |
| 13 | \( 1 + (0.993 + 0.116i)T^{2} \) |
| 17 | \( 1 + (-0.109 + 0.0397i)T + (0.766 - 0.642i)T^{2} \) |
| 19 | \( 1 + (-0.539 - 0.196i)T + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.973 + 0.230i)T^{2} \) |
| 29 | \( 1 + (-0.597 - 0.802i)T^{2} \) |
| 31 | \( 1 + (0.286 - 0.957i)T^{2} \) |
| 37 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 41 | \( 1 + (1.62 + 1.06i)T + (0.396 + 0.918i)T^{2} \) |
| 43 | \( 1 + (1.62 - 0.385i)T + (0.893 - 0.448i)T^{2} \) |
| 47 | \( 1 + (0.286 + 0.957i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.393 - 1.31i)T + (-0.835 + 0.549i)T^{2} \) |
| 61 | \( 1 + (0.686 - 0.727i)T^{2} \) |
| 67 | \( 1 + (1.77 - 0.891i)T + (0.597 - 0.802i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.310 - 1.76i)T + (-0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (-0.396 + 0.918i)T^{2} \) |
| 83 | \( 1 + (1.28 - 0.841i)T + (0.396 - 0.918i)T^{2} \) |
| 89 | \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (0.819 + 0.868i)T + (-0.0581 + 0.998i)T^{2} \) |
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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
−10.18279637291063025365927915627, −9.996377236206970391058707720124, −8.864544284636239313845996398488, −8.295334321166935172529876851481, −7.35547564091018071915714978991, −6.80444039504943201603201544976, −5.54809311855952725675199140986, −4.32834115717736976024749010718, −2.80059202278079678370494757819, −1.62119570828251856248690417829,
1.63438328148246247318544547598, 2.98171378947512783007168620984, 3.64758248090820471619726462084, 5.04351899938963909760166427321, 6.62401873579710065623808000357, 7.58455753829830154652018205865, 8.302909250005651854433082013501, 9.017926989531276790830733026409, 9.786379636895055593710768410146, 10.52028314302954469050532094733