| L(s) = 1 | + (−1.39 + 1.43i)2-s + (0.447 − 0.367i)3-s + (−0.118 − 3.99i)4-s + (−2.47 − 0.750i)5-s + (−0.0964 + 1.15i)6-s + (0.987 + 4.96i)7-s + (5.90 + 5.39i)8-s + (−1.69 + 8.49i)9-s + (4.52 − 2.50i)10-s + (0.355 + 3.60i)11-s + (−1.52 − 1.74i)12-s + (6.50 + 21.4i)13-s + (−8.49 − 5.49i)14-s + (−1.38 + 0.572i)15-s + (−15.9 + 0.948i)16-s + (9.43 − 22.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.717i)2-s + (0.149 − 0.122i)3-s + (−0.0296 − 0.999i)4-s + (−0.494 − 0.150i)5-s + (−0.0160 + 0.192i)6-s + (0.141 + 0.708i)7-s + (0.737 + 0.674i)8-s + (−0.187 + 0.944i)9-s + (0.452 − 0.250i)10-s + (0.0323 + 0.328i)11-s + (−0.126 − 0.145i)12-s + (0.500 + 1.65i)13-s + (−0.606 − 0.392i)14-s + (−0.0922 + 0.0381i)15-s + (−0.998 + 0.0593i)16-s + (0.554 − 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.439716 + 0.698761i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.439716 + 0.698761i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 1.43i)T \) |
| good | 3 | \( 1 + (-0.447 + 0.367i)T + (1.75 - 8.82i)T^{2} \) |
| 5 | \( 1 + (2.47 + 0.750i)T + (20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (-0.987 - 4.96i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-0.355 - 3.60i)T + (-118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (-6.50 - 21.4i)T + (-140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-9.43 + 22.7i)T + (-204. - 204. i)T^{2} \) |
| 19 | \( 1 + (13.6 - 25.6i)T + (-200. - 300. i)T^{2} \) |
| 23 | \( 1 + (6.73 - 4.50i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (-0.212 + 2.15i)T + (-824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-36.6 - 36.6i)T + 961iT^{2} \) |
| 37 | \( 1 + (10.2 + 19.1i)T + (-760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-63.1 + 42.1i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (35.7 + 29.3i)T + (360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (15.2 - 36.8i)T + (-1.56e3 - 1.56e3i)T^{2} \) |
| 53 | \( 1 + (-1.07 - 10.9i)T + (-2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (55.6 + 16.8i)T + (2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (-12.5 + 10.3i)T + (725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (-58.9 - 71.8i)T + (-875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-50.6 + 10.0i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (60.4 + 12.0i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-6.16 - 14.8i)T + (-4.41e3 + 4.41e3i)T^{2} \) |
| 83 | \( 1 + (90.7 + 48.4i)T + (3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (4.77 - 7.14i)T + (-3.03e3 - 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-107. + 107. i)T - 9.40e3iT^{2} \) |
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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
−13.97480998648137068962516094577, −12.20858899736098129286109600566, −11.34969690654108106291208710005, −10.09536668903650801829353994287, −8.957565273706434161179886845872, −8.147915055511004525589530551042, −7.12269179086342573175940557956, −5.80130697734093057172979608667, −4.50057352418870531654426145112, −1.97434408181173846794421123543,
0.72427202728610119290294221824, 3.10982378053286614760218589478, 4.11586404255317724949617096472, 6.29952887530135761312113342136, 7.80487414851477906888834113919, 8.460185941057147762488905024851, 9.805161633439550402358277771096, 10.70031778426211797143091198846, 11.50529082449105244684981867414, 12.68335118351656806111931364545
