L(s) = 1 | + (−0.5 − 0.866i)3-s + (2.13 − 3.70i)5-s + (1.5 + 2.17i)7-s + (−0.499 + 0.866i)9-s + (−2.13 − 3.70i)11-s − 1.27·13-s − 4.27·15-s + (2 + 3.46i)17-s + (0.637 − 1.10i)19-s + (1.13 − 2.38i)21-s + (2 − 3.46i)23-s + (−6.63 − 11.4i)25-s + 0.999·27-s − 2.27·29-s + (−0.5 − 0.866i)31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.955 − 1.65i)5-s + (0.566 + 0.823i)7-s + (−0.166 + 0.288i)9-s + (−0.644 − 1.11i)11-s − 0.353·13-s − 1.10·15-s + (0.485 + 0.840i)17-s + (0.146 − 0.253i)19-s + (0.248 − 0.521i)21-s + (0.417 − 0.722i)23-s + (−1.32 − 2.29i)25-s + 0.192·27-s − 0.422·29-s + (−0.0898 − 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05131 - 0.911179i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05131 - 0.911179i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.5 - 2.17i)T \) |
good | 5 | \( 1 + (-2.13 + 3.70i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.13 + 3.70i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.637 + 1.10i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.27T + 29T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.63 - 4.56i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 7.27T + 43T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.862 + 1.49i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.13 - 5.43i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.63 - 6.30i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + (1.63 + 2.83i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.77 - 3.07i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 0.274T + 83T^{2} \) |
| 89 | \( 1 + (2.27 - 3.94i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−11.51317479163195447605038051501, −10.44333375533891504036756965945, −9.227614784754786840831208184393, −8.582145083761049158782860866095, −7.83008210436565779230548107869, −6.01533140034928252569747787192, −5.57128198500792941915531630452, −4.63217650865892246754143326071, −2.45882049048464737088415181622, −1.11471273770735904449800676994,
2.14477394519740374842223426050, 3.42452853812675334369098323117, 4.86776942251035515082418012651, 5.86279335594290059257680124061, 7.18681853954463542255624733924, 7.49059981103733252696925178363, 9.488728496285496435116844264274, 9.993697363967175716383257733913, 10.77820448811673137663390704828, 11.34927966234222508008437280018