L(s) = 1 | + (−1.58 − 0.917i)2-s + (0.682 + 1.18i)4-s + 3.80·5-s + (−2.43 − 1.03i)7-s + 1.16i·8-s + (−6.04 − 3.49i)10-s − 1.16i·11-s + (−4.79 − 2.77i)13-s + (2.91 + 3.88i)14-s + (2.43 − 4.21i)16-s + (1.58 − 2.75i)17-s + (−0.546 + 0.315i)19-s + (2.59 + 4.50i)20-s + (−1.06 + 1.85i)22-s − 1.76i·23-s + ⋯ |
L(s) = 1 | + (−1.12 − 0.648i)2-s + (0.341 + 0.590i)4-s + 1.70·5-s + (−0.919 − 0.392i)7-s + 0.412i·8-s + (−1.91 − 1.10i)10-s − 0.351i·11-s + (−1.33 − 0.768i)13-s + (0.778 + 1.03i)14-s + (0.608 − 1.05i)16-s + (0.385 − 0.667i)17-s + (−0.125 + 0.0724i)19-s + (0.580 + 1.00i)20-s + (−0.227 + 0.394i)22-s − 0.367i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.360385 - 0.704851i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.360385 - 0.704851i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.43 + 1.03i)T \) |
good | 2 | \( 1 + (1.58 + 0.917i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 3.80T + 5T^{2} \) |
| 11 | \( 1 + 1.16iT - 11T^{2} \) |
| 13 | \( 1 + (4.79 + 2.77i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.58 + 2.75i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.546 - 0.315i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 1.76iT - 23T^{2} \) |
| 29 | \( 1 + (-4.18 + 2.41i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.70 + 2.13i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.11 + 3.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.28 + 3.95i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.61 + 6.26i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.27 - 2.20i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0627 - 0.0362i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.49 + 6.04i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.00 + 4.04i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.54 + 4.41i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.76iT - 71T^{2} \) |
| 73 | \( 1 + (-4.84 - 2.79i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.54 - 14.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.08 - 8.80i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.77 - 10.0i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.596 + 0.344i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.18847215890019711790428671416, −9.696183780988217957306274243049, −9.160745723141093607086438803823, −8.015213303863084818532661130109, −6.91170006312194026241277956652, −5.87675195438714719558167171912, −5.01870558546105164520557500413, −2.98306136267249859094617014626, −2.21559845984140918453685917343, −0.64759303471121258745501010350,
1.63102344620754218889415868199, 2.92248372835269572133708893660, 4.78061433086636029815566531691, 6.03215902022224197661981550463, 6.54941781632695240377795313382, 7.41687514857795689445976367804, 8.709179783634068231293321712054, 9.300995786890100745717134878674, 10.06300383411785824681742749411, 10.22630738310165129175119825214