L(s) = 1 | + (−1.5 − 0.866i)3-s − 7-s + (1.5 + 2.59i)9-s + (1.5 − 0.866i)13-s + (4 − 1.73i)19-s + (1.5 + 0.866i)21-s + (−2.5 − 4.33i)25-s − 5.19i·27-s + 1.73i·31-s − 12.1i·37-s − 3·39-s + (2.5 − 4.33i)43-s − 6·49-s + (−7.5 − 0.866i)57-s + (−6.5 − 11.2i)61-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)3-s − 0.377·7-s + (0.5 + 0.866i)9-s + (0.416 − 0.240i)13-s + (0.917 − 0.397i)19-s + (0.327 + 0.188i)21-s + (−0.5 − 0.866i)25-s − 0.999i·27-s + 0.311i·31-s − 1.99i·37-s − 0.480·39-s + (0.381 − 0.660i)43-s − 0.857·49-s + (−0.993 − 0.114i)57-s + (−0.832 − 1.44i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0170 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0170 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.657019 - 0.668330i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.657019 - 0.668330i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 19 | \( 1 + (-4 + 1.73i)T \) |
good | 5 | \( 1 + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-1.5 + 0.866i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 12.1iT - 37T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.5 + 4.33i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-13.5 + 7.79i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (10.5 + 6.06i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12 - 6.92i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.00311657342454386641840395283, −9.121033248371343188949767000476, −8.004075526395432552469405186401, −7.26394257254266819427114318211, −6.37311003893507907746973752706, −5.65117602599111564984811374913, −4.71747194763352177561202472590, −3.48685352687846023996732429693, −2.06899686290975617007166025818, −0.55974054361500708227279987436,
1.27241638092739712913031359346, 3.13379200516269274789181400945, 4.08046024924639346782630873004, 5.11273958969276149091409984587, 5.94013360447493335798086731790, 6.71175807179879474444745423776, 7.67801626416124460551599245233, 8.807284673552640923989318948932, 9.736228114929307421622786169791, 10.13897162808116930670349604205