| L(s) = 1 | + (−1.23 − 0.690i)2-s + (0.540 − 0.841i)3-s + (1.04 + 1.70i)4-s + (0.282 + 0.128i)5-s + (−1.24 + 0.664i)6-s + (−2.03 − 0.597i)7-s + (−0.114 − 2.82i)8-s + (−0.415 − 0.909i)9-s + (−0.259 − 0.354i)10-s + (−2.44 − 2.12i)11-s + (1.99 + 0.0412i)12-s + (0.746 + 2.54i)13-s + (2.09 + 2.14i)14-s + (0.261 − 0.167i)15-s + (−1.81 + 3.56i)16-s + (−0.439 − 3.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.872 − 0.488i)2-s + (0.312 − 0.485i)3-s + (0.523 + 0.852i)4-s + (0.126 + 0.0576i)5-s + (−0.509 + 0.271i)6-s + (−0.769 − 0.225i)7-s + (−0.0404 − 0.999i)8-s + (−0.138 − 0.303i)9-s + (−0.0820 − 0.111i)10-s + (−0.738 − 0.639i)11-s + (0.577 + 0.0119i)12-s + (0.206 + 0.704i)13-s + (0.561 + 0.572i)14-s + (0.0674 − 0.0433i)15-s + (−0.452 + 0.891i)16-s + (−0.106 − 0.741i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0382386 - 0.507131i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0382386 - 0.507131i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 0.690i)T \) |
| 3 | \( 1 + (-0.540 + 0.841i)T \) |
| 23 | \( 1 + (2.97 + 3.76i)T \) |
| good | 5 | \( 1 + (-0.282 - 0.128i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (2.03 + 0.597i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (2.44 + 2.12i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.746 - 2.54i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.439 + 3.05i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (5.09 + 0.732i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-3.10 + 0.446i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-7.94 + 5.10i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (4.16 - 1.90i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-2.96 + 6.49i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (6.68 - 10.4i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + (0.793 - 2.70i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (3.65 + 12.4i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-4.26 - 6.63i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-2.86 + 2.48i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-4.09 - 4.72i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.30 - 9.08i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-1.58 + 0.465i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-3.67 + 1.68i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-2.05 - 1.31i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-5.20 + 11.3i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.18226416535215844082758824625, −9.613563581986395769371309168544, −8.471149083462790152385066394552, −8.045201879214915831467773429019, −6.71438495428982431934916952360, −6.30642290480105510572206584412, −4.36051188570547741863542523480, −3.08140299362443475307237093839, −2.16660471488032485260618625132, −0.34552059429718952156512491870,
1.97455878052182985933924437065, 3.30266160212510896432539339751, 4.85137188316406736568535716857, 5.87472645043929887452973351549, 6.72185571052691166658658440290, 7.929442598883733222222646377054, 8.482564570642909390253578768403, 9.516687801551171994670362291825, 10.18889489254365877345596704974, 10.66363987104991411428515290888
