L(s) = 1 | + (0.5 + 0.866i)2-s + (−1.34 + 1.09i)3-s + (−0.499 + 0.866i)4-s + (2.44 − 1.41i)5-s + (−1.61 − 0.613i)6-s + (2.60 − 0.449i)7-s − 0.999·8-s + (0.599 − 2.93i)9-s + (2.44 + 1.41i)10-s + (2.97 − 5.15i)11-s + (−0.278 − 1.70i)12-s + (−3.07 − 1.87i)13-s + (1.69 + 2.03i)14-s + (−1.73 + 4.58i)15-s + (−0.5 − 0.866i)16-s + (2.83 − 4.91i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.774 + 0.632i)3-s + (−0.249 + 0.433i)4-s + (1.09 − 0.632i)5-s + (−0.661 − 0.250i)6-s + (0.985 − 0.169i)7-s − 0.353·8-s + (0.199 − 0.979i)9-s + (0.774 + 0.447i)10-s + (0.897 − 1.55i)11-s + (−0.0802 − 0.493i)12-s + (−0.854 − 0.520i)13-s + (0.452 + 0.543i)14-s + (−0.448 + 1.18i)15-s + (−0.125 − 0.216i)16-s + (0.688 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.69944 + 0.375780i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.69944 + 0.375780i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (1.34 - 1.09i)T \) |
| 7 | \( 1 + (-2.60 + 0.449i)T \) |
| 13 | \( 1 + (3.07 + 1.87i)T \) |
good | 5 | \( 1 + (-2.44 + 1.41i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.97 + 5.15i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.83 + 4.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.972 - 1.68i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.43 - 1.40i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.0658iT - 29T^{2} \) |
| 31 | \( 1 + (4.39 - 7.61i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.160 - 0.0928i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.04iT - 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 + (6.77 - 3.91i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.77 + 2.18i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-11.6 - 6.71i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (12.8 - 7.39i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.912 + 0.527i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.16T + 71T^{2} \) |
| 73 | \( 1 + (-0.140 + 0.242i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.18 - 7.24i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.940iT - 83T^{2} \) |
| 89 | \( 1 + (4.61 - 2.66i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 9.60T + 97T^{2} \) |
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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
−10.93054216566779820845347032594, −9.839886662012931895773658130161, −9.187190467315781784927155323829, −8.248570878418647219944639481606, −7.05938075961429659541281289645, −5.82954064994806554920401520770, −5.44402180846171963418171808523, −4.62528418932493040715770683924, −3.30006131143940664595550913071, −1.14235637846141174613716788274,
1.77554261580359027961502109024, 2.12769994129273800671831079788, 4.22043830775535463846380983882, 5.15138399526102420824383795197, 6.07638496326952267359021805186, 6.91597518545790249858218368929, 7.85685246881886248963622275042, 9.384731221630194954837330354905, 10.04413206695953539835860788166, 10.83180396042370694559291027746