L(s) = 1 | + (−0.382 − 0.923i)2-s + (0.182 + 0.917i)3-s + (−0.707 + 0.707i)4-s + (−0.566 − 2.16i)5-s + (0.778 − 0.519i)6-s + (1.62 − 1.08i)7-s + (0.923 + 0.382i)8-s + (1.96 − 0.812i)9-s + (−1.78 + 1.35i)10-s + (−2.02 − 3.02i)11-s + (−0.778 − 0.519i)12-s + 6.15·13-s + (−1.62 − 1.08i)14-s + (1.88 − 0.915i)15-s − i·16-s + (−0.742 + 4.05i)17-s + ⋯ |
L(s) = 1 | + (−0.270 − 0.653i)2-s + (0.105 + 0.529i)3-s + (−0.353 + 0.353i)4-s + (−0.253 − 0.967i)5-s + (0.317 − 0.212i)6-s + (0.614 − 0.410i)7-s + (0.326 + 0.135i)8-s + (0.654 − 0.270i)9-s + (−0.563 + 0.427i)10-s + (−0.609 − 0.911i)11-s + (−0.224 − 0.150i)12-s + 1.70·13-s + (−0.434 − 0.290i)14-s + (0.485 − 0.236i)15-s − 0.250i·16-s + (−0.180 + 0.983i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.896725 - 0.556960i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.896725 - 0.556960i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.382 + 0.923i)T \) |
| 5 | \( 1 + (0.566 + 2.16i)T \) |
| 17 | \( 1 + (0.742 - 4.05i)T \) |
good | 3 | \( 1 + (-0.182 - 0.917i)T + (-2.77 + 1.14i)T^{2} \) |
| 7 | \( 1 + (-1.62 + 1.08i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (2.02 + 3.02i)T + (-4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 - 6.15T + 13T^{2} \) |
| 19 | \( 1 + (2.19 + 0.909i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (3.52 + 0.700i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (5.01 - 0.996i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (0.311 - 0.466i)T + (-11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (-4.49 + 0.893i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-4.93 - 0.981i)T + (37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (3.01 - 7.27i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 - 12.6iT - 47T^{2} \) |
| 53 | \( 1 + (-6.86 + 2.84i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (5.12 + 12.3i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-0.856 + 4.30i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-5.21 - 5.21i)T + 67iT^{2} \) |
| 71 | \( 1 + (-4.65 - 3.11i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (1.98 + 1.32i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (11.1 - 7.44i)T + (30.2 - 72.9i)T^{2} \) |
| 83 | \( 1 + (4.02 + 9.70i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-1.02 + 1.02i)T - 89iT^{2} \) |
| 97 | \( 1 + (13.1 + 8.76i)T + (37.1 + 89.6i)T^{2} \) |
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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
−12.78260064807445608593220147824, −11.22766395785023270580625410893, −10.83833059195743116088800883132, −9.582003914527649493270797554911, −8.555711541316931160639332450038, −7.956025526137160666721051468844, −5.98689768281907775194092086966, −4.47298084337615316221726636170, −3.69143760788615285511395949375, −1.32378200971959358858570210426,
2.07794411326717826578391494957, 4.13243027576021186741908753441, 5.69019736742569468842208275384, 6.89435658113380494579233094723, 7.63223496039183656851476063721, 8.551526980712028267602715191924, 9.966319028724921606580890313373, 10.87201008568915853077976675675, 11.91365389865048252846534040176, 13.23661817067739869243518096850