L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s + (−0.5 + 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)11-s + 0.999·12-s + (2.5 + 2.59i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s − 0.999·18-s + (−2.5 + 4.33i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.204 − 0.353i)6-s + (−0.188 + 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.452 + 0.783i)11-s + 0.288·12-s + (0.693 + 0.720i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s − 0.235·18-s + (−0.573 + 0.993i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01854 + 1.00556i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01854 + 1.00556i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-2.5 - 2.59i)T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 16T + 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 - 18T + 83T^{2} \) |
| 89 | \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8 + 13.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.27467675444554506627348569976, −10.02605263889822091944870710653, −9.122340129898367408359917653256, −8.206461721416714192763360892477, −7.20688672234468133435500442623, −6.50643086392239607624689136621, −5.62708270514104307497978292971, −4.56427042930626092281714506410, −3.36605741552866436779474637925, −1.69814671503214991208220847581,
0.826149236098139701427696371742, 2.78155526401293814983481276927, 3.82168485186125513243235329820, 4.70971873196188448076294038605, 5.93282311916951321499209389756, 6.57202235152663544342250571000, 8.218760440740126732519333153441, 8.862927880834299839115338554495, 10.17921788330878460074369981724, 10.43665446237303050423048562125