L(s) = 1 | + (−1.52 − 0.819i)3-s + (−0.769 − 0.206i)5-s + (2.17 − 3.76i)7-s + (1.65 + 2.50i)9-s + (−3.93 + 1.05i)11-s + (1.69 + 0.454i)13-s + (1.00 + 0.945i)15-s − 6.68i·17-s + (0.708 + 0.708i)19-s + (−6.40 + 3.96i)21-s + (−3.88 + 2.24i)23-s + (−3.78 − 2.18i)25-s + (−0.475 − 5.17i)27-s + (−3.98 + 1.06i)29-s + (−4.94 + 2.85i)31-s + ⋯ |
L(s) = 1 | + (−0.880 − 0.473i)3-s + (−0.344 − 0.0922i)5-s + (0.822 − 1.42i)7-s + (0.551 + 0.833i)9-s + (−1.18 + 0.318i)11-s + (0.470 + 0.126i)13-s + (0.259 + 0.244i)15-s − 1.62i·17-s + (0.162 + 0.162i)19-s + (−1.39 + 0.865i)21-s + (−0.809 + 0.467i)23-s + (−0.756 − 0.436i)25-s + (−0.0914 − 0.995i)27-s + (−0.740 + 0.198i)29-s + (−0.887 + 0.512i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.885 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.151146 - 0.614569i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.151146 - 0.614569i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.52 + 0.819i)T \) |
good | 5 | \( 1 + (0.769 + 0.206i)T + (4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (-2.17 + 3.76i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.93 - 1.05i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-1.69 - 0.454i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + 6.68iT - 17T^{2} \) |
| 19 | \( 1 + (-0.708 - 0.708i)T + 19iT^{2} \) |
| 23 | \( 1 + (3.88 - 2.24i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.98 - 1.06i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (4.94 - 2.85i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.51 + 1.51i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.36 + 2.36i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.30 + 8.60i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-1.23 + 2.13i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.68 - 1.68i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.269 + 1.00i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (0.528 + 1.97i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-2.14 + 8.01i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 8.05iT - 71T^{2} \) |
| 73 | \( 1 - 9.73iT - 73T^{2} \) |
| 79 | \( 1 + (-11.9 - 6.91i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.817 + 3.05i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 1.71T + 89T^{2} \) |
| 97 | \( 1 + (-3.66 + 6.34i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.57603393853914428789525810544, −9.776419634217108164911204537054, −8.236173405910266866188196451092, −7.43849801187027278958167517638, −7.08500536660792232984999423942, −5.57018086458132306588415986573, −4.85533450377210063953875219651, −3.81399406431882313566508537259, −1.90058736570164439244684488690, −0.39141915748779937528085295734,
1.94249524455860362749134806707, 3.52963130929359156650294846479, 4.72780355993984171174062968395, 5.69007340397167782270727101277, 6.11551984381695616847850876503, 7.75697632437955082424490235316, 8.376805206816033661801743223768, 9.392109882680421747360633025030, 10.46586152770104025539558197808, 11.11528376268866272046637097416