L(s) = 1 | + (0.5 − 0.866i)3-s + (−2 − 3.46i)5-s + (2.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−3 + 5.19i)11-s − 5·13-s − 3.99·15-s + (−1 + 1.73i)17-s + (−0.5 − 0.866i)19-s + (2 − 1.73i)21-s + (−3 − 5.19i)23-s + (−5.49 + 9.52i)25-s − 0.999·27-s + (−1.5 + 2.59i)31-s + (3 + 5.19i)33-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.894 − 1.54i)5-s + (0.944 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.904 + 1.56i)11-s − 1.38·13-s − 1.03·15-s + (−0.242 + 0.420i)17-s + (−0.114 − 0.198i)19-s + (0.436 − 0.377i)21-s + (−0.625 − 1.08i)23-s + (−1.09 + 1.90i)25-s − 0.192·27-s + (−0.269 + 0.466i)31-s + (0.522 + 0.904i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 5 | \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (2 + 3.46i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.888894369879400097993493848140, −8.137627364123077207530094041446, −7.73430200867319171148577948312, −6.94175021694050638782766532680, −5.39448727056468430255608451844, −4.74595426399781404789886099900, −4.27341296974478721417591623125, −2.49305908448567367030785356109, −1.61916457107724997243687807398, 0,
2.35840850751405595125872558675, 3.16042618390013366449468201205, 3.97631658247478653323370928207, 5.01923809027054233052667864290, 5.96573570114839498057923853526, 7.18554149881087138515494660794, 7.73981743416146513036747746327, 8.223373703662598902568952040242, 9.419382354141753284742995101688