L(s) = 1 | + (3.05 + 1.76i)2-s + (4.23 + 7.34i)4-s + (−0.533 + 0.923i)5-s + 0.477·7-s + 15.8i·8-s + (−3.26 + 1.88i)10-s + 11.1·11-s + (−12.9 + 7.49i)13-s + (1.45 + 0.842i)14-s + (−10.9 + 19.0i)16-s + (6.11 − 10.5i)17-s + (−6.52 − 17.8i)19-s − 9.03·20-s + (34.1 + 19.7i)22-s + (−20.7 − 35.9i)23-s + ⋯ |
L(s) = 1 | + (1.52 + 0.883i)2-s + (1.05 + 1.83i)4-s + (−0.106 + 0.184i)5-s + 0.0681·7-s + 1.97i·8-s + (−0.326 + 0.188i)10-s + 1.01·11-s + (−0.998 + 0.576i)13-s + (0.104 + 0.0602i)14-s + (−0.686 + 1.18i)16-s + (0.359 − 0.623i)17-s + (−0.343 − 0.939i)19-s − 0.451·20-s + (1.55 + 0.896i)22-s + (−0.901 − 1.56i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0752 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0752 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.38475 + 2.21167i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.38475 + 2.21167i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 + (6.52 + 17.8i)T \) |
good | 2 | \( 1 + (-3.05 - 1.76i)T + (2 + 3.46i)T^{2} \) |
| 5 | \( 1 + (0.533 - 0.923i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 - 0.477T + 49T^{2} \) |
| 11 | \( 1 - 11.1T + 121T^{2} \) |
| 13 | \( 1 + (12.9 - 7.49i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-6.11 + 10.5i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (20.7 + 35.9i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-30.5 + 17.6i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 15.8iT - 961T^{2} \) |
| 37 | \( 1 + 24.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-5.83 - 3.37i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (31.6 - 54.8i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (20.2 + 35.0i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (26.3 - 15.2i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-69.1 - 39.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (35.8 + 62.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-7.62 + 4.40i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-42.2 - 24.4i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (56.7 - 98.3i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-104. - 60.3i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 65.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + (133. - 76.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (83.0 + 47.9i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.85779396983088886822758878485, −12.09169331568405685298591842387, −11.30604362630941493385148390855, −9.683433439404635297119233399492, −8.295374986613224253739944189770, −7.00105050116020131892372748738, −6.48567954033792575814636304195, −5.02183892025749073444846638835, −4.19481637671908257042329553979, −2.73491459731614340969944701637,
1.67087352388340331862219089436, 3.28428691420936871192945215184, 4.33581256701005924747688623182, 5.47694688333101171389857247221, 6.52420465599889697651304507444, 8.096105771523402113012550864843, 9.756114051529023828084166096806, 10.53132334123534203489148791913, 11.83362275136858106661015131784, 12.17664227569489443543533581589