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 | \( 1 + (13.1 - 8.76i)T + (37.1 - 89.6i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.23661817067739869243518096850, −11.91365389865048252846534040176, −10.87201008568915853077976675675, −9.966319028724921606580890313373, −8.551526980712028267602715191924, −7.63223496039183656851476063721, −6.89435658113380494579233094723, −5.69019736742569468842208275384, −4.13243027576021186741908753441, −2.07794411326717826578391494957,
1.32378200971959358858570210426, 3.69143760788615285511395949375, 4.47298084337615316221726636170, 5.98689768281907775194092086966, 7.956025526137160666721051468844, 8.555711541316931160639332450038, 9.582003914527649493270797554911, 10.83833059195743116088800883132, 11.22766395785023270580625410893, 12.78260064807445608593220147824