L(s) = 1 | + (−0.686 + 0.727i)2-s + (0.597 + 0.802i)3-s + (−0.0581 − 0.998i)4-s + (−0.993 − 0.116i)6-s + (0.766 + 0.642i)8-s + (−0.286 + 0.957i)9-s + (−0.0460 + 0.106i)11-s + (0.766 − 0.642i)12-s + (−0.993 + 0.116i)16-s + (0.337 + 1.91i)17-s + (−0.500 − 0.866i)18-s + (0.137 − 0.780i)19-s + (−0.0460 − 0.106i)22-s + (−0.0581 + 0.998i)24-s + (0.973 + 0.230i)25-s + ⋯ |
L(s) = 1 | + (−0.686 + 0.727i)2-s + (0.597 + 0.802i)3-s + (−0.0581 − 0.998i)4-s + (−0.993 − 0.116i)6-s + (0.766 + 0.642i)8-s + (−0.286 + 0.957i)9-s + (−0.0460 + 0.106i)11-s + (0.766 − 0.642i)12-s + (−0.993 + 0.116i)16-s + (0.337 + 1.91i)17-s + (−0.500 − 0.866i)18-s + (0.137 − 0.780i)19-s + (−0.0460 − 0.106i)22-s + (−0.0581 + 0.998i)24-s + (0.973 + 0.230i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7914801250\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7914801250\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.686 - 0.727i)T \) |
| 3 | \( 1 + (-0.597 - 0.802i)T \) |
good | 5 | \( 1 + (-0.973 - 0.230i)T^{2} \) |
| 7 | \( 1 + (-0.597 + 0.802i)T^{2} \) |
| 11 | \( 1 + (0.0460 - 0.106i)T + (-0.686 - 0.727i)T^{2} \) |
| 13 | \( 1 + (-0.893 + 0.448i)T^{2} \) |
| 17 | \( 1 + (-0.337 - 1.91i)T + (-0.939 + 0.342i)T^{2} \) |
| 19 | \( 1 + (-0.137 + 0.780i)T + (-0.939 - 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.597 - 0.802i)T^{2} \) |
| 29 | \( 1 + (0.835 - 0.549i)T^{2} \) |
| 31 | \( 1 + (-0.396 + 0.918i)T^{2} \) |
| 37 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 41 | \( 1 + (0.819 + 0.868i)T + (-0.0581 + 0.998i)T^{2} \) |
| 43 | \( 1 + (0.819 + 1.10i)T + (-0.286 + 0.957i)T^{2} \) |
| 47 | \( 1 + (-0.396 - 0.918i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.786 + 1.82i)T + (-0.686 + 0.727i)T^{2} \) |
| 61 | \( 1 + (0.993 + 0.116i)T^{2} \) |
| 67 | \( 1 + (0.512 - 1.71i)T + (-0.835 - 0.549i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (0.439 + 0.368i)T + (0.173 + 0.984i)T^{2} \) |
| 79 | \( 1 + (0.0581 + 0.998i)T^{2} \) |
| 83 | \( 1 + (-1.28 + 1.36i)T + (-0.0581 - 0.998i)T^{2} \) |
| 89 | \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (-1.65 + 0.193i)T + (0.973 - 0.230i)T^{2} \) |
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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
−10.46612903538974459009079675638, −10.21693579602008771493739847712, −9.019360033771816685679335269459, −8.595491441131312720328377392701, −7.70834799402477478340171557210, −6.71221249139882609614951667793, −5.59463353390529508232630017436, −4.71629305523329904731433404693, −3.52396726306414145871935528108, −1.94171306769174556875580755088,
1.19277327915257926647068187499, 2.60807273551117587801918806412, 3.36276082695495321447019622914, 4.80832192965778853144308670945, 6.37490551991027724729027819247, 7.34508908679926416971536262260, 7.933872298852693664507230723008, 8.900408160436850079694509048485, 9.516072226693409049906744010507, 10.42261711646253025309794381910