L(s) = 1 | + (−1.17 + 0.788i)2-s + (0.757 − 1.85i)4-s − i·5-s + 2.12·7-s + (0.569 + 2.77i)8-s + (0.788 + 1.17i)10-s + 5.48i·11-s + 4.91i·13-s + (−2.49 + 1.67i)14-s + (−2.85 − 2.80i)16-s + 0.235·17-s − 5.23i·19-s + (−1.85 − 0.757i)20-s + (−4.31 − 6.43i)22-s − 7.42·23-s + ⋯ |
L(s) = 1 | + (−0.830 + 0.557i)2-s + (0.378 − 0.925i)4-s − 0.447i·5-s + 0.804·7-s + (0.201 + 0.979i)8-s + (0.249 + 0.371i)10-s + 1.65i·11-s + 1.36i·13-s + (−0.667 + 0.448i)14-s + (−0.712 − 0.701i)16-s + 0.0571·17-s − 1.20i·19-s + (−0.413 − 0.169i)20-s + (−0.920 − 1.37i)22-s − 1.54·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9757616791\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9757616791\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.17 - 0.788i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 - 2.12T + 7T^{2} \) |
| 11 | \( 1 - 5.48iT - 11T^{2} \) |
| 13 | \( 1 - 4.91iT - 13T^{2} \) |
| 17 | \( 1 - 0.235T + 17T^{2} \) |
| 19 | \( 1 + 5.23iT - 19T^{2} \) |
| 23 | \( 1 + 7.42T + 23T^{2} \) |
| 29 | \( 1 - 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 5.05T + 31T^{2} \) |
| 37 | \( 1 - 2.27iT - 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 - 9.90iT - 43T^{2} \) |
| 47 | \( 1 - 8.17T + 47T^{2} \) |
| 53 | \( 1 - 10.6iT - 53T^{2} \) |
| 59 | \( 1 + 0.702iT - 59T^{2} \) |
| 61 | \( 1 + 0.319iT - 61T^{2} \) |
| 67 | \( 1 - 2.70iT - 67T^{2} \) |
| 71 | \( 1 + 7.18T + 71T^{2} \) |
| 73 | \( 1 - 9.66T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 8.53iT - 83T^{2} \) |
| 89 | \( 1 - 9.90T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−9.843757589149614037143267325972, −9.261672881157976772459673308915, −8.491375707220335073478422895640, −7.61992610224552774602069598464, −6.97814782876473809083763476911, −6.08974249104795237505316895652, −4.75007586188034935457502052392, −4.51754740193626043129162151303, −2.28210139787628271056554328113, −1.44728973368172890973086778124,
0.59357354700352900801364461825, 2.03889731188371147864148933417, 3.19840010926763325079635065586, 3.95424589853710499960096585974, 5.55093842293756371535513396246, 6.25280532028417123931833129213, 7.58091306929786885130379602114, 8.183403372868340367299241662648, 8.569653785930226695953636198353, 9.939414565500704189975931398960