L(s) = 1 | + 3.41·3-s + 1.54i·5-s + 2.65·9-s + 4.48·11-s − 1.54i·13-s + 5.29i·15-s − 23.6·17-s − 24.8·19-s + 35.2i·23-s + 22.5·25-s − 21.6·27-s + 22.4i·29-s + 46.7i·31-s + 15.3·33-s + 58.5i·37-s + ⋯ |
L(s) = 1 | + 1.13·3-s + 0.309i·5-s + 0.295·9-s + 0.407·11-s − 0.119i·13-s + 0.352i·15-s − 1.39·17-s − 1.30·19-s + 1.53i·23-s + 0.903·25-s − 0.802·27-s + 0.774i·29-s + 1.50i·31-s + 0.464·33-s + 1.58i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.916406153\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.916406153\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 3.41T + 9T^{2} \) |
| 5 | \( 1 - 1.54iT - 25T^{2} \) |
| 11 | \( 1 - 4.48T + 121T^{2} \) |
| 13 | \( 1 + 1.54iT - 169T^{2} \) |
| 17 | \( 1 + 23.6T + 289T^{2} \) |
| 19 | \( 1 + 24.8T + 361T^{2} \) |
| 23 | \( 1 - 35.2iT - 529T^{2} \) |
| 29 | \( 1 - 22.4iT - 841T^{2} \) |
| 31 | \( 1 - 46.7iT - 961T^{2} \) |
| 37 | \( 1 - 58.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 26.9T + 1.68e3T^{2} \) |
| 43 | \( 1 - 17.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 36.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 97.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 61.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 37.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 33.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 69.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 38.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 3.61T + 6.88e3T^{2} \) |
| 89 | \( 1 + 44.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + 96.1T + 9.40e3T^{2} \) |
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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
−9.264099461124304131166335463784, −8.671477013134521619614539575245, −8.155886573888047961550863361249, −7.01853935020823688856698065934, −6.55741365776454680153292312798, −5.30426045057631724313799538485, −4.26247114350214120415264585625, −3.36980081905970546345539246319, −2.56843144724443263535100519102, −1.53645884194557283523959050890,
0.40927760673123102181992078825, 2.13116396006314271017607466200, 2.60901009795583957786016839770, 4.08856029755866181897582988341, 4.37792425187638811890489654832, 5.85549538213598833498271527292, 6.62745589523231803450380510285, 7.55976444826415371261914423303, 8.472712043749645561228096629000, 8.871497904485816330287467817226