L(s) = 1 | + (0.146 + 0.146i)3-s + (−3.68 − 3.68i)5-s + 9.66·7-s − 8.95i·9-s + (5.51 − 5.51i)11-s + (6.27 − 6.27i)13-s − 1.07i·15-s − 6.78·17-s + (13.5 + 13.5i)19-s + (1.41 + 1.41i)21-s − 17.0·23-s + 2.17i·25-s + (2.62 − 2.62i)27-s + (−4.85 + 4.85i)29-s − 5.25i·31-s + ⋯ |
L(s) = 1 | + (0.0487 + 0.0487i)3-s + (−0.737 − 0.737i)5-s + 1.38·7-s − 0.995i·9-s + (0.501 − 0.501i)11-s + (0.482 − 0.482i)13-s − 0.0719i·15-s − 0.399·17-s + (0.711 + 0.711i)19-s + (0.0673 + 0.0673i)21-s − 0.742·23-s + 0.0868i·25-s + (0.0973 − 0.0973i)27-s + (−0.167 + 0.167i)29-s − 0.169i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.28483 - 0.610715i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28483 - 0.610715i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-0.146 - 0.146i)T + 9iT^{2} \) |
| 5 | \( 1 + (3.68 + 3.68i)T + 25iT^{2} \) |
| 7 | \( 1 - 9.66T + 49T^{2} \) |
| 11 | \( 1 + (-5.51 + 5.51i)T - 121iT^{2} \) |
| 13 | \( 1 + (-6.27 + 6.27i)T - 169iT^{2} \) |
| 17 | \( 1 + 6.78T + 289T^{2} \) |
| 19 | \( 1 + (-13.5 - 13.5i)T + 361iT^{2} \) |
| 23 | \( 1 + 17.0T + 529T^{2} \) |
| 29 | \( 1 + (4.85 - 4.85i)T - 841iT^{2} \) |
| 31 | \( 1 + 5.25iT - 961T^{2} \) |
| 37 | \( 1 + (-18.1 - 18.1i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 48.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (54.5 - 54.5i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 40.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (10.8 + 10.8i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-50.8 + 50.8i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (-17.0 + 17.0i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (-22.9 - 22.9i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 51.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 78.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 108. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-57.3 - 57.3i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 44.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 112.T + 9.40e3T^{2} \) |
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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
−12.80978142334865857411281985997, −11.76212633651133948346070175260, −11.28352406155003716726192569975, −9.704769410609953027441049197993, −8.453863103014025218110663542876, −7.943865604055658519035849366271, −6.23294661261759384540442188834, −4.81195408157694058549468912775, −3.66162520057796070896004407721, −1.14666983541597035126335982429,
2.02279808850000456988517517984, 3.97927758473003112946790116109, 5.17522856817213202663390405637, 6.97906298916505510026264133683, 7.77843680313576712498394008786, 8.828233926513025312699682303178, 10.42423034135515169326788489772, 11.32239588029299951092380265210, 11.83512446379618725767202580789, 13.48199774613989454729768346724