| L(s) = 1 | − 0.792·3-s − i·5-s + (−0.792 + 2.52i)7-s − 2.37·9-s − 0.792i·11-s − 5.37i·13-s + 0.792i·15-s − 3.37i·17-s − 3.46·19-s + (0.627 − 2i)21-s − 1.87i·23-s − 25-s + 4.25·27-s − 5.37·29-s − 8.51·31-s + ⋯ |
| L(s) = 1 | − 0.457·3-s − 0.447i·5-s + (−0.299 + 0.954i)7-s − 0.790·9-s − 0.238i·11-s − 1.49i·13-s + 0.204i·15-s − 0.817i·17-s − 0.794·19-s + (0.136 − 0.436i)21-s − 0.391i·23-s − 0.200·25-s + 0.819·27-s − 0.997·29-s − 1.52·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.169920 - 0.436103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.169920 - 0.436103i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (0.792 - 2.52i)T \) |
| good | 3 | \( 1 + 0.792T + 3T^{2} \) |
| 11 | \( 1 + 0.792iT - 11T^{2} \) |
| 13 | \( 1 + 5.37iT - 13T^{2} \) |
| 17 | \( 1 + 3.37iT - 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 1.87iT - 23T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 + 8.51T + 31T^{2} \) |
| 37 | \( 1 + 0.744T + 37T^{2} \) |
| 41 | \( 1 + 2.74iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 + 0.744iT - 61T^{2} \) |
| 67 | \( 1 + 6.63iT - 67T^{2} \) |
| 71 | \( 1 + 6.63iT - 71T^{2} \) |
| 73 | \( 1 + 2.74iT - 73T^{2} \) |
| 79 | \( 1 - 14.0iT - 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 17.4iT - 89T^{2} \) |
| 97 | \( 1 - 2.11iT - 97T^{2} \) |
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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
−10.57578461076778577541810602524, −9.434721269723958226591478630519, −8.695652891948642316824601665815, −7.930151318954172597273155756198, −6.61373160657578985081089906216, −5.53342314040045577268640099210, −5.25078498343270238929489119488, −3.52893174465180249957119305977, −2.38385921909605538809543723148, −0.26648329701262441283466365805,
1.88859062289562034901248085527, 3.50996311768675077783846342037, 4.40111132149176130117710613804, 5.73370548972512298226335501174, 6.61652669075350722218364347920, 7.29588079125236800740730075628, 8.483918716735134463174846186941, 9.436590504712309059629825225000, 10.36554968605600058372184290340, 11.15221922504530068838915624627
