| L(s) = 1 | + (−1.37 + 0.323i)2-s + (1.79 − 0.890i)4-s + (−0.470 + 1.29i)5-s + (1.57 + 0.277i)7-s + (−2.17 + 1.80i)8-s + (0.229 − 1.93i)10-s + (3.66 − 1.33i)11-s + (−5.10 + 4.28i)13-s + (−2.25 + 0.126i)14-s + (2.41 − 3.18i)16-s + (2.32 − 1.34i)17-s + (3.15 + 1.82i)19-s + (0.308 + 2.73i)20-s + (−4.61 + 3.02i)22-s + (0.644 + 3.65i)23-s + ⋯ |
| L(s) = 1 | + (−0.973 + 0.228i)2-s + (0.895 − 0.445i)4-s + (−0.210 + 0.577i)5-s + (0.595 + 0.104i)7-s + (−0.769 + 0.638i)8-s + (0.0725 − 0.610i)10-s + (1.10 − 0.402i)11-s + (−1.41 + 1.18i)13-s + (−0.603 + 0.0339i)14-s + (0.603 − 0.797i)16-s + (0.564 − 0.326i)17-s + (0.724 + 0.418i)19-s + (0.0688 + 0.610i)20-s + (−0.984 + 0.644i)22-s + (0.134 + 0.761i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.753460 + 0.453442i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.753460 + 0.453442i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 0.323i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.470 - 1.29i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.57 - 0.277i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.66 + 1.33i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (5.10 - 4.28i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.32 + 1.34i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.15 - 1.82i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.644 - 3.65i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.98 - 2.36i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.45 + 0.255i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.27 - 2.20i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.30 - 7.51i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.27 - 3.49i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.901 + 5.11i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 3.96iT - 53T^{2} \) |
| 59 | \( 1 + (-0.666 - 0.242i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.969 + 5.49i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (10.4 + 12.4i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (7.76 + 13.4i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.04 + 7.00i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.37 + 9.98i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.344 + 0.289i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (0.994 + 0.574i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.00 - 3.27i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.65741040094698543920686620460, −10.87118086815145555923572119627, −9.612277662366181480112879957193, −9.201578303985236208431668405915, −7.84094801974331574035721168381, −7.20011638562268061890906661668, −6.23636996291462355510457837880, −4.89146486372325477860508908553, −3.16443879677514387051540811817, −1.59363393562257277858796033148,
0.961593450281762820784700411311, 2.61728030303476639846707536632, 4.22992350205151348461689036937, 5.53909279988283652746094144208, 7.02038002210004962625907143454, 7.76981120342296777146674732692, 8.671066494249793019466304624534, 9.594244319633608577852175434165, 10.36673575915871911013175597626, 11.42391115867035639771959625353
