| L(s) = 1 | + (0.157 + 0.0533i)3-s + (0.226 − 3.45i)5-s + (−2.23 + 1.41i)7-s + (−2.35 − 1.80i)9-s + (−0.172 − 0.150i)11-s + (−0.727 − 0.727i)13-s + (0.220 − 0.531i)15-s + (−1.71 − 3.75i)17-s + (−4.50 + 0.593i)19-s + (−0.427 + 0.102i)21-s + (0.610 + 1.79i)23-s + (−6.94 − 0.913i)25-s + (−0.550 − 0.824i)27-s + (5.75 + 3.84i)29-s + (0.648 − 1.91i)31-s + ⋯ |
| L(s) = 1 | + (0.0907 + 0.0308i)3-s + (0.101 − 1.54i)5-s + (−0.845 + 0.534i)7-s + (−0.786 − 0.603i)9-s + (−0.0518 − 0.0454i)11-s + (−0.201 − 0.201i)13-s + (0.0568 − 0.137i)15-s + (−0.415 − 0.909i)17-s + (−1.03 + 0.136i)19-s + (−0.0931 + 0.0224i)21-s + (0.127 + 0.375i)23-s + (−1.38 − 0.182i)25-s + (−0.106 − 0.158i)27-s + (1.06 + 0.714i)29-s + (0.116 − 0.343i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.324668 - 0.766279i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.324668 - 0.766279i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.23 - 1.41i)T \) |
| 17 | \( 1 + (1.71 + 3.75i)T \) |
| good | 3 | \( 1 + (-0.157 - 0.0533i)T + (2.38 + 1.82i)T^{2} \) |
| 5 | \( 1 + (-0.226 + 3.45i)T + (-4.95 - 0.652i)T^{2} \) |
| 11 | \( 1 + (0.172 + 0.150i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.727 + 0.727i)T + 13iT^{2} \) |
| 19 | \( 1 + (4.50 - 0.593i)T + (18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (-0.610 - 1.79i)T + (-18.2 + 14.0i)T^{2} \) |
| 29 | \( 1 + (-5.75 - 3.84i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (-0.648 + 1.91i)T + (-24.5 - 18.8i)T^{2} \) |
| 37 | \( 1 + (6.47 + 7.38i)T + (-4.82 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.38 + 2.93i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-3.92 + 1.62i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-2.88 + 10.7i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.49 - 1.91i)T + (13.7 - 51.1i)T^{2} \) |
| 59 | \( 1 + (-0.682 + 5.18i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (0.994 - 2.01i)T + (-37.1 - 48.3i)T^{2} \) |
| 67 | \( 1 + (-11.4 - 6.60i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (14.4 - 2.87i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-4.52 + 2.22i)T + (44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (-3.59 + 1.22i)T + (62.6 - 48.0i)T^{2} \) |
| 83 | \( 1 + (-8.14 - 3.37i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-12.6 - 3.39i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (1.25 - 1.88i)T + (-37.1 - 89.6i)T^{2} \) |
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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
−10.61009288951738021317021704051, −9.404741998792557195736263118232, −8.982564050945800310290181339660, −8.313778939495126479257196506550, −6.88954976137687731173925863949, −5.80318505056013891440918453491, −5.06453912186101970179060580522, −3.79547177128508495442856395333, −2.42248565213164015195737348395, −0.47160249301242296579025040287,
2.38576830127327596420466395862, 3.21599671762086486876903306722, 4.47247077404859785560910487288, 6.17818753217597928185773158027, 6.54719793885745287992160043820, 7.60552437681012926261204253241, 8.575263835589019350512257407474, 9.789647818040284131542526501806, 10.66934825580836542457179991619, 10.90780235253739160953817375217
