L(s) = 1 | + (1.14 − 1.97i)2-s + (−1.61 − 2.79i)4-s + (−0.5 + 0.866i)5-s + 3.50·7-s − 2.79·8-s + (1.14 + 1.97i)10-s + 4.50·11-s + (2.5 + 4.33i)13-s + (4.00 − 6.94i)14-s + (0.0316 − 0.0547i)16-s + (0.0793 − 0.137i)17-s + (−4.26 + 0.920i)19-s + 3.22·20-s + (5.14 − 8.91i)22-s + (0.579 + 1.00i)23-s + ⋯ |
L(s) = 1 | + (0.807 − 1.39i)2-s + (−0.805 − 1.39i)4-s + (−0.223 + 0.387i)5-s + 1.32·7-s − 0.987·8-s + (0.361 + 0.625i)10-s + 1.35·11-s + (0.693 + 1.20i)13-s + (1.07 − 1.85i)14-s + (0.00790 − 0.0136i)16-s + (0.0192 − 0.0333i)17-s + (−0.977 + 0.211i)19-s + 0.720·20-s + (1.09 − 1.90i)22-s + (0.120 + 0.209i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.85499 - 2.04609i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.85499 - 2.04609i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (4.26 - 0.920i)T \) |
good | 2 | \( 1 + (-1.14 + 1.97i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 11 | \( 1 - 4.50T + 11T^{2} \) |
| 13 | \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.0793 + 0.137i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.579 - 1.00i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.75 + 3.03i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.28T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + (-3.03 + 5.26i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.67 + 2.89i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.53 - 2.65i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.87 + 4.97i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.53 + 2.65i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.436 - 0.756i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.22 - 7.31i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (8.11 - 14.0i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.57 + 6.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.06 + 8.76i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.85T + 83T^{2} \) |
| 89 | \( 1 + (0.556 + 0.963i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.809 - 1.40i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.39354919981124765160343514981, −9.238950611962673681696853742890, −8.564532177398196825210068669359, −7.31570119157122832581456215509, −6.30271380838113754286792217197, −5.13559357468523693885187638810, −4.11036052168669984667063970235, −3.76632719722354185379734238878, −2.14093075211920399384381005937, −1.44696843022817216828331123756,
1.47748021961720739342696918665, 3.55506685921991911433188923144, 4.40579250954443567283355052516, 5.15893207963018382639306160436, 5.99836827841205679684707179520, 6.87145421752408642689763502480, 7.79182665501498912104753063854, 8.435836517989934679562575119270, 9.020288027350677198385784275450, 10.58397754737383480373750546392