| L(s) = 1 | + (−0.479 + 1.97i)2-s + (−0.327 + 0.945i)3-s + (−1.89 − 0.978i)4-s + (3.28 + 1.31i)5-s + (−1.71 − 1.09i)6-s + (0.889 − 2.49i)7-s + (0.179 − 0.207i)8-s + (−0.786 − 0.618i)9-s + (−4.17 + 5.86i)10-s + (0.938 + 3.86i)11-s + (1.54 − 1.47i)12-s + (−0.232 − 0.509i)13-s + (4.49 + 2.95i)14-s + (−2.31 + 2.67i)15-s + (−2.15 − 3.02i)16-s + (−0.0693 + 1.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.338 + 1.39i)2-s + (−0.188 + 0.545i)3-s + (−0.948 − 0.489i)4-s + (1.47 + 0.588i)5-s + (−0.698 − 0.448i)6-s + (0.336 − 0.941i)7-s + (0.0635 − 0.0733i)8-s + (−0.262 − 0.206i)9-s + (−1.32 + 1.85i)10-s + (0.282 + 1.16i)11-s + (0.446 − 0.425i)12-s + (−0.0645 − 0.141i)13-s + (1.20 + 0.789i)14-s + (−0.598 + 0.691i)15-s + (−0.538 − 0.755i)16-s + (−0.0168 + 0.352i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.148425 + 1.37064i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.148425 + 1.37064i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.327 - 0.945i)T \) |
| 7 | \( 1 + (-0.889 + 2.49i)T \) |
| 23 | \( 1 + (-0.416 - 4.77i)T \) |
| good | 2 | \( 1 + (0.479 - 1.97i)T + (-1.77 - 0.916i)T^{2} \) |
| 5 | \( 1 + (-3.28 - 1.31i)T + (3.61 + 3.45i)T^{2} \) |
| 11 | \( 1 + (-0.938 - 3.86i)T + (-9.77 + 5.04i)T^{2} \) |
| 13 | \( 1 + (0.232 + 0.509i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (0.0693 - 1.45i)T + (-16.9 - 1.61i)T^{2} \) |
| 19 | \( 1 + (-0.219 - 4.59i)T + (-18.9 + 1.80i)T^{2} \) |
| 29 | \( 1 + (5.64 + 3.62i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.49 - 0.866i)T + (28.7 + 11.5i)T^{2} \) |
| 37 | \( 1 + (6.20 + 4.87i)T + (8.72 + 35.9i)T^{2} \) |
| 41 | \( 1 + (0.713 + 4.96i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (0.474 + 0.547i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-4.46 + 7.72i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.48 + 0.810i)T + (52.0 - 10.0i)T^{2} \) |
| 59 | \( 1 + (0.382 - 0.536i)T + (-19.2 - 55.7i)T^{2} \) |
| 61 | \( 1 + (-0.383 - 1.10i)T + (-47.9 + 37.7i)T^{2} \) |
| 67 | \( 1 + (-3.14 - 3.00i)T + (3.18 + 66.9i)T^{2} \) |
| 71 | \( 1 + (-5.70 + 1.67i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (0.927 + 0.478i)T + (42.3 + 59.4i)T^{2} \) |
| 79 | \( 1 + (-6.39 - 0.610i)T + (77.5 + 14.9i)T^{2} \) |
| 83 | \( 1 + (0.499 - 3.47i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (13.4 - 2.59i)T + (82.6 - 33.0i)T^{2} \) |
| 97 | \( 1 + (-0.828 - 5.76i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.05444514022367976990809717159, −9.995432770983763591325606274886, −9.814708904698828767286306682273, −8.665557166601428008419931996474, −7.45650961118038103467416939227, −6.88722248446514720045834074834, −5.86980668559855799127316298158, −5.23417876992715825734514558854, −3.89110963958977248407559507955, −1.98424592069183499716453831842,
1.02236884462492966370076331325, 2.12002098040246635386172284736, 2.95637664269295951280651897575, 4.83782781373010661773132213929, 5.82017145885728780485346015693, 6.62707787160208043186480211049, 8.471247083294454690218136700435, 8.976514675930341801524580334895, 9.664461229030741775022804762383, 10.72911661870056402602443227771
