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
−10.13897162808116930670349604205, −9.736228114929307421622786169791, −8.807284673552640923989318948932, −7.67801626416124460551599245233, −6.71175807179879474444745423776, −5.94013360447493335798086731790, −5.11273958969276149091409984587, −4.08046024924639346782630873004, −3.13379200516269274789181400945, −1.27241638092739712913031359346,
0.55974054361500708227279987436, 2.06899686290975617007166025818, 3.48685352687846023996732429693, 4.71747194763352177561202472590, 5.65117602599111564984811374913, 6.37311003893507907746973752706, 7.26394257254266819427114318211, 8.004075526395432552469405186401, 9.121033248371343188949767000476, 10.00311657342454386641840395283