L(s) = 1 | + (−0.5 − 0.866i)5-s + 2i·7-s − 2i·11-s + (−1.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (1.73 + 4i)19-s + (0.866 + 0.5i)23-s + (2 − 3.46i)25-s + (1.5 + 0.866i)29-s − 3.46·31-s + (1.73 − i)35-s + 3.46i·37-s + (−4.5 + 2.59i)41-s + (7.79 − 4.5i)43-s + (9.52 + 5.5i)47-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s + 0.755i·7-s − 0.603i·11-s + (−0.416 − 0.240i)13-s + (0.121 + 0.210i)17-s + (0.397 + 0.917i)19-s + (0.180 + 0.104i)23-s + (0.400 − 0.692i)25-s + (0.278 + 0.160i)29-s − 0.622·31-s + (0.292 − 0.169i)35-s + 0.569i·37-s + (−0.702 + 0.405i)41-s + (1.18 − 0.686i)43-s + (1.38 + 0.802i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.632692141\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.632692141\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-1.73 - 4i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 0.866i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 3.46iT - 37T^{2} \) |
| 41 | \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.79 + 4.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-9.52 - 5.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.5 + 4.33i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.33 - 7.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.59 + 4.5i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.79 - 13.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.866 + 1.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + (-13.5 - 7.79i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (13.5 - 7.79i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.769225004677905628196884913470, −8.210541967857736156764399547082, −7.48470239466300338638086910517, −6.48426871394514074481996854472, −5.68681027450689196882613956309, −5.10901088863053932715570219792, −4.07648523359913160380116721928, −3.16823521032454024210876061760, −2.20476987571863987148729585213, −0.890298257165407281855597979795,
0.71719748984312550499089244316, 2.10880101523515606431501660447, 3.11184663614998544006087200731, 4.03684102087668396967238404704, 4.79829773390585926101805701546, 5.64056174685023744387040549497, 6.84496340477391857733181661092, 7.17408267018293847329822615871, 7.80552511459070888497457933254, 8.931037779697432124777504963780