| L(s) = 1 | + (1.20 + 0.747i)2-s + (0.881 + 1.79i)4-s + (−3.32 + 0.585i)5-s + (−1.72 + 2.05i)7-s + (−0.285 + 2.81i)8-s + (−4.42 − 1.78i)10-s + (−0.506 + 2.87i)11-s + (0.552 + 0.201i)13-s + (−3.61 + 1.17i)14-s + (−2.44 + 3.16i)16-s + (6.36 − 3.67i)17-s + (2.82 + 1.63i)19-s + (−3.97 − 5.44i)20-s + (−2.75 + 3.07i)22-s + (−0.365 + 0.306i)23-s + ⋯ |
| L(s) = 1 | + (0.848 + 0.528i)2-s + (0.440 + 0.897i)4-s + (−1.48 + 0.262i)5-s + (−0.652 + 0.777i)7-s + (−0.100 + 0.994i)8-s + (−1.39 − 0.563i)10-s + (−0.152 + 0.866i)11-s + (0.153 + 0.0557i)13-s + (−0.965 + 0.314i)14-s + (−0.611 + 0.791i)16-s + (1.54 − 0.890i)17-s + (0.648 + 0.374i)19-s + (−0.889 − 1.21i)20-s + (−0.587 + 0.654i)22-s + (−0.0761 + 0.0638i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.616989 + 1.29725i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.616989 + 1.29725i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.20 - 0.747i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.32 - 0.585i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.72 - 2.05i)T + (-1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.506 - 2.87i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.552 - 0.201i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-6.36 + 3.67i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.82 - 1.63i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.365 - 0.306i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (1.49 + 4.10i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.73 - 4.45i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (1.59 + 2.76i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.09 + 3.01i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.79 + 0.316i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.940 - 0.789i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 7.37iT - 53T^{2} \) |
| 59 | \( 1 + (-0.718 - 4.07i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.421 - 0.353i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.46 - 9.51i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.616 - 1.06i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.18 + 2.04i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.33 + 9.16i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-16.8 + 6.15i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-4.81 - 2.77i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.83 + 10.4i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−12.08768392647804422336059790526, −11.55161711348756091757460888578, −10.16137444820912371695535293727, −8.909972301475106391672465624096, −7.69302409319504528850066403811, −7.32598552795820394362122439699, −6.02678796794271811623676510573, −4.93325857468311484801730486602, −3.74133212769757425324767628199, −2.85694754911512684061310723916,
0.818460052118756829782994785642, 3.33094334902166371969185585468, 3.73677174399260058153715694743, 5.02695112131585205685608639690, 6.25870681191796730790645459747, 7.40226032224062429173069294432, 8.279493546448529224343423849755, 9.727837463335192407515165018529, 10.63182338949338544186080240548, 11.44243736890230226675955994949
