L(s) = 1 | + (−0.776 − 1.34i)2-s + (0.5 − 0.866i)3-s + (−0.204 + 0.355i)4-s − 1.55·6-s + (−2.60 + 0.478i)7-s − 2.46·8-s + (−0.499 − 0.866i)9-s + (−2.21 + 3.83i)11-s + (0.204 + 0.355i)12-s − 1.73·13-s + (2.66 + 3.12i)14-s + (2.32 + 4.02i)16-s + (−1.36 + 2.36i)17-s + (−0.776 + 1.34i)18-s + (0.152 + 0.264i)19-s + ⋯ |
L(s) = 1 | + (−0.548 − 0.950i)2-s + (0.288 − 0.499i)3-s + (−0.102 + 0.177i)4-s − 0.633·6-s + (−0.983 + 0.180i)7-s − 0.872·8-s + (−0.166 − 0.288i)9-s + (−0.667 + 1.15i)11-s + (0.0591 + 0.102i)12-s − 0.480·13-s + (0.711 + 0.835i)14-s + (0.581 + 1.00i)16-s + (−0.331 + 0.573i)17-s + (−0.182 + 0.316i)18-s + (0.0350 + 0.0606i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0639562 + 0.0720182i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0639562 + 0.0720182i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.60 - 0.478i)T \) |
good | 2 | \( 1 + (0.776 + 1.34i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (2.21 - 3.83i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 + (1.36 - 2.36i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.152 - 0.264i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.51 + 6.08i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 + (-2.64 + 4.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.83 + 3.18i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.71T + 41T^{2} \) |
| 43 | \( 1 - 9.71T + 43T^{2} \) |
| 47 | \( 1 + (-0.908 - 1.57i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.857 - 1.48i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.571 + 0.989i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.77 + 8.27i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.19 - 7.26i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + (6.41 - 11.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.35 + 5.81i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 5.09T + 83T^{2} \) |
| 89 | \( 1 + (2.03 + 3.52i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2.87T + 97T^{2} \) |
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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
−10.10361322576104016834977184982, −9.576699496704525665965394397845, −8.688683159292168589442587278454, −7.60308331796093946159307682982, −6.61530736792496704181316438406, −5.68791098143105775883429264988, −4.06999311059652771109222045440, −2.71114158039884935537272635104, −1.99327323852734034382137584623, −0.05894336131456572209763637020,
2.81474293417123297050899886060, 3.64772531163757781052421907379, 5.30183488732725081706104096138, 6.09853921553533646769399588308, 7.16030273674771106064391612234, 7.87737468049116625337622815631, 8.862042171321727744071377732640, 9.470568970985353546949189608578, 10.32570981324247574124171537359, 11.37301536649559223125073451407