| L(s) = 1 | + (1.40 + 1.01i)3-s + (1.34 + 3.68i)5-s + (0.499 + 2.83i)7-s + (0.956 + 2.84i)9-s + (0.996 − 2.73i)11-s + (2.55 − 3.04i)13-s + (−1.84 + 6.54i)15-s + (2.48 − 4.30i)17-s + (1.78 − 1.02i)19-s + (−2.16 + 4.49i)21-s + (−0.165 + 0.936i)23-s + (−7.96 + 6.68i)25-s + (−1.52 + 4.96i)27-s + (−5.18 − 6.17i)29-s + (−0.331 + 1.88i)31-s + ⋯ |
| L(s) = 1 | + (0.812 + 0.583i)3-s + (0.600 + 1.64i)5-s + (0.188 + 1.07i)7-s + (0.318 + 0.947i)9-s + (0.300 − 0.825i)11-s + (0.709 − 0.845i)13-s + (−0.475 + 1.68i)15-s + (0.603 − 1.04i)17-s + (0.408 − 0.235i)19-s + (−0.472 + 0.980i)21-s + (−0.0344 + 0.195i)23-s + (−1.59 + 1.33i)25-s + (−0.294 + 0.955i)27-s + (−0.962 − 1.14i)29-s + (−0.0595 + 0.337i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0683 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0683 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.65670 + 1.77402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65670 + 1.77402i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.40 - 1.01i)T \) |
| good | 5 | \( 1 + (-1.34 - 3.68i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.499 - 2.83i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.996 + 2.73i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.55 + 3.04i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.48 + 4.30i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.78 + 1.02i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.165 - 0.936i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (5.18 + 6.17i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.331 - 1.88i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (8.31 + 4.79i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.581 - 0.487i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.14 - 5.88i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.932 + 5.28i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 3.54iT - 53T^{2} \) |
| 59 | \( 1 + (2.13 + 5.86i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-3.22 + 0.568i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.40 + 5.25i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (3.83 - 6.64i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.46 - 6.00i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.205 - 0.172i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.91 + 9.42i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (4.11 + 7.12i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.06 - 1.84i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.24679707584291261667726334865, −9.614964547207358746017300482201, −8.805358526194503353879217342063, −7.921835855530774731580876341589, −7.02858117067308913822900472977, −5.89587181258361060102248693838, −5.32320087602050898753241465652, −3.54356615254890296139693946797, −3.04130135506662894657310848011, −2.10423340412423860098255602713,
1.28300853772345328264722467401, 1.72706594273887037311967701881, 3.70378563698084586211245423492, 4.37108971528901043547154768967, 5.52089348810102647045335158663, 6.63631748807136294655932553864, 7.50417608472304337032632203661, 8.377997661629673692282144213688, 9.013806400349397629637892279814, 9.694382880551390152427932106509
