L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.499 − 0.866i)6-s − 7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)10-s + 3.77·11-s + 0.999·12-s + (2.38 − 4.13i)13-s + (−0.5 − 0.866i)14-s + (−0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + (0.204 − 0.353i)6-s − 0.377·7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 − 0.273i)10-s + 1.13·11-s + 0.288·12-s + (0.661 − 1.14i)13-s + (−0.133 − 0.231i)14-s + (−0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44099 - 0.301765i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44099 - 0.301765i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-2.88 + 3.26i)T \) |
good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 + (-2.38 + 4.13i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-1.88 + 3.26i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-1.88 - 3.26i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.38 + 7.59i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.88 - 8.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.77 - 11.7i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.38 - 2.40i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.38 + 9.32i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.77 - 6.53i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.61 - 2.79i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.15 + 8.93i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (8.65 - 14.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.82540974983481333471424884927, −9.642127164153768272063868732509, −8.694965034724487515105278609053, −7.952607136391824461827559761158, −6.90386483072044435145287749766, −6.21572377067093524337183460088, −5.26059112806970958328214933517, −4.15240873729066086379028745080, −2.95744198429218754653464790257, −0.897036902763835848930715897183,
1.49379574705798438540401070417, 3.27889352658067559263446694536, 3.92043005706454492913084453699, 5.02278075545527508646029796417, 6.25294014105870489654570349695, 6.86685834437280169404905797608, 8.387877012469351659656048114021, 9.395698081598537866653099355433, 9.892622392004703816488616237977, 11.04774909708136299323012228940