L(s) = 1 | + (−1.40 + 0.191i)2-s + (1.66 − 0.493i)3-s + (1.92 − 0.535i)4-s + (1.09 + 1.89i)5-s + (−2.23 + 1.00i)6-s + (−0.451 + 2.60i)7-s + (−2.59 + 1.11i)8-s + (2.51 − 1.63i)9-s + (−1.89 − 2.44i)10-s + (−1.45 − 0.837i)11-s + (2.93 − 1.84i)12-s − 1.56i·13-s + (0.134 − 3.73i)14-s + (2.74 + 2.60i)15-s + (3.42 − 2.06i)16-s + (−0.278 − 0.160i)17-s + ⋯ |
L(s) = 1 | + (−0.990 + 0.135i)2-s + (0.958 − 0.285i)3-s + (0.963 − 0.267i)4-s + (0.488 + 0.845i)5-s + (−0.911 + 0.412i)6-s + (−0.170 + 0.985i)7-s + (−0.918 + 0.395i)8-s + (0.837 − 0.546i)9-s + (−0.598 − 0.772i)10-s + (−0.437 − 0.252i)11-s + (0.847 − 0.531i)12-s − 0.432i·13-s + (0.0360 − 0.999i)14-s + (0.709 + 0.671i)15-s + (0.856 − 0.516i)16-s + (−0.0676 − 0.0390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03986 + 0.243923i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03986 + 0.243923i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 - 0.191i)T \) |
| 3 | \( 1 + (-1.66 + 0.493i)T \) |
| 7 | \( 1 + (0.451 - 2.60i)T \) |
good | 5 | \( 1 + (-1.09 - 1.89i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.45 + 0.837i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.56iT - 13T^{2} \) |
| 17 | \( 1 + (0.278 + 0.160i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.87 - 4.97i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.26 + 5.65i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7.04T + 29T^{2} \) |
| 31 | \( 1 + (7.76 + 4.48i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.946 + 0.546i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.44iT - 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + (2.25 + 3.90i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.42 - 7.67i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.37 - 3.10i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.80 + 2.19i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.716 + 1.24i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.37T + 71T^{2} \) |
| 73 | \( 1 + (4.49 - 7.77i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.58 - 2.07i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.06iT - 83T^{2} \) |
| 89 | \( 1 + (-4.57 + 2.64i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.23T + 97T^{2} \) |
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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
−12.78383731979940718722213965819, −11.83414525131187620937622445338, −10.41104323168853364742458663864, −9.838352787553420748733133234882, −8.679809299593629668195851636660, −7.957293794478883098143463024899, −6.75181232843014596426745392505, −5.79118253206917112785127236324, −3.11712107354151336769277963077, −2.15233386550754000059974311409,
1.59522635659105395110869557634, 3.30964842663194320103253453849, 4.91465296100028961726091296011, 6.85610521708067659416526357753, 7.77165894805617437671425469963, 8.824070439160240220228404010050, 9.611122899419814644809220417785, 10.29087142620368347271005467148, 11.50268732798604875868145908659, 12.92180982768892066912189461932