L(s) = 1 | + (0.5 − 0.866i)3-s + (−1.66 + 2.87i)5-s − 2.71i·7-s + (−0.499 − 0.866i)9-s − 0.985i·11-s + (4.33 − 2.50i)13-s + (1.66 + 2.87i)15-s + (−2.51 + 4.35i)17-s + (0.193 − 4.35i)19-s + (−2.35 − 1.35i)21-s + (3.68 − 2.12i)23-s + (−3.01 − 5.22i)25-s − 0.999·27-s + (5.83 − 3.36i)29-s + 2.32·31-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.742 + 1.28i)5-s − 1.02i·7-s + (−0.166 − 0.288i)9-s − 0.297i·11-s + (1.20 − 0.694i)13-s + (0.428 + 0.742i)15-s + (−0.609 + 1.05i)17-s + (0.0443 − 0.999i)19-s + (−0.513 − 0.296i)21-s + (0.769 − 0.444i)23-s + (−0.602 − 1.04i)25-s − 0.192·27-s + (1.08 − 0.625i)29-s + 0.416·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29302 - 0.701064i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29302 - 0.701064i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.193 + 4.35i)T \) |
good | 5 | \( 1 + (1.66 - 2.87i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 2.71iT - 7T^{2} \) |
| 11 | \( 1 + 0.985iT - 11T^{2} \) |
| 13 | \( 1 + (-4.33 + 2.50i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.51 - 4.35i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.68 + 2.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.83 + 3.36i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.32T + 31T^{2} \) |
| 37 | \( 1 + 8.27iT - 37T^{2} \) |
| 41 | \( 1 + (9.96 + 5.75i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.48 - 5.47i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.41 + 3.70i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.14 + 2.97i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.66 - 2.87i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.62 - 6.28i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.67 + 11.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.19 + 3.79i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.52 - 4.37i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.48 - 9.49i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8.70iT - 83T^{2} \) |
| 89 | \( 1 + (6.10 - 3.52i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.12 + 4.11i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.37495237600157443550263146428, −8.860576738377096264223904821855, −8.199649636068270282829852916653, −7.28054032842554347925304499634, −6.78134481025544307209084541395, −5.92579317664484770853800802862, −4.25077285515649905344936071940, −3.52634388155989796220498972618, −2.57931515889451875789821064976, −0.77343962202013966290894783796,
1.34863762532301722798065003016, 2.89657813202755012730545361237, 4.09318778451537032527630377959, 4.81450083628724642924597460536, 5.65022787069198418781300276425, 6.83495918578915257977255162055, 8.056348228142345729140561684854, 8.773743635187947895727800148443, 9.028238777152784505424698520226, 10.05134694694065298534571465609