| L(s) = 1 | + (1.41 − i)3-s − 1.41i·5-s + i·7-s + (1.00 − 2.82i)9-s + 1.41·11-s + 2·13-s + (−1.41 − 2.00i)15-s − 7.07i·17-s + 4i·19-s + (1 + 1.41i)21-s − 7.07·23-s + 2.99·25-s + (−1.41 − 5.00i)27-s + 2.82i·29-s + 6i·31-s + ⋯ |
| L(s) = 1 | + (0.816 − 0.577i)3-s − 0.632i·5-s + 0.377i·7-s + (0.333 − 0.942i)9-s + 0.426·11-s + 0.554·13-s + (−0.365 − 0.516i)15-s − 1.71i·17-s + 0.917i·19-s + (0.218 + 0.308i)21-s − 1.47·23-s + 0.599·25-s + (−0.272 − 0.962i)27-s + 0.525i·29-s + 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54147 - 0.797923i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54147 - 0.797923i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 + i)T \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 7.07iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 11.3iT - 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 7.07T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 12iT - 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 9.89iT - 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−11.84253930630199945612207119040, −10.32770641690859401095504476516, −9.228603216731903836396765225115, −8.705393159899842781090626948296, −7.72598716346095390093213703429, −6.71800158128003087366850180932, −5.54876582500712198584767803174, −4.15968064781942441795551230241, −2.86871493481765532390313929576, −1.36095168638243533824219021592,
2.09325074645280039147583441777, 3.54971813439294422320974746924, 4.27530916424374366714358493111, 5.88625886774990327113080535099, 6.98747729218547172607141553437, 8.085895987375839302562937428026, 8.837020454948251689036398153114, 9.980027140494724818264259400272, 10.58471053077945084326230716516, 11.44349607349479309237910184864
