| L(s) = 1 | + (−0.978 + 0.207i)2-s + (0.104 + 0.994i)3-s + (0.913 − 0.406i)4-s + (1.06 − 1.18i)5-s + (−0.309 − 0.951i)6-s + (−2.48 − 0.912i)7-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (−0.794 + 1.37i)10-s + (−1.41 − 3.00i)11-s + (0.5 + 0.866i)12-s + (1.65 − 5.09i)13-s + (2.61 + 0.376i)14-s + (1.28 + 0.933i)15-s + (0.669 − 0.743i)16-s + (−7.14 − 1.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 + 0.147i)2-s + (0.0603 + 0.574i)3-s + (0.456 − 0.203i)4-s + (0.475 − 0.528i)5-s + (−0.126 − 0.388i)6-s + (−0.938 − 0.345i)7-s + (−0.286 + 0.207i)8-s + (−0.326 + 0.0693i)9-s + (−0.251 + 0.435i)10-s + (−0.425 − 0.904i)11-s + (0.144 + 0.250i)12-s + (0.459 − 1.41i)13-s + (0.699 + 0.100i)14-s + (0.331 + 0.241i)15-s + (0.167 − 0.185i)16-s + (−1.73 − 0.368i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0696 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0696 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.506860 - 0.472708i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.506860 - 0.472708i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 - 0.207i)T \) |
| 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 7 | \( 1 + (2.48 + 0.912i)T \) |
| 11 | \( 1 + (1.41 + 3.00i)T \) |
| good | 5 | \( 1 + (-1.06 + 1.18i)T + (-0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (-1.65 + 5.09i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (7.14 + 1.51i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.841 - 0.374i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.477 - 0.826i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.30 + 3.12i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.69 - 1.88i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-1.14 + 10.8i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (-2.76 + 2.01i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + (2.70 + 1.20i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (7.15 + 7.95i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-3.93 + 1.75i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-6.76 + 7.51i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (6.94 - 12.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.23 - 6.89i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.59 - 0.710i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (10.2 - 2.18i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-1.40 - 4.31i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-8.07 - 13.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.23 + 6.88i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.77872159927082806220931057193, −9.784653798332840358788821545766, −9.131356435571907971076401587465, −8.375869069244735399193054508806, −7.26739275863467415632521519568, −6.04106976659848540529889821940, −5.38074519941622170375119113965, −3.81440623423029526165403796155, −2.61848526521243310947204272594, −0.50225039606529266715861946382,
1.90234512535588401337379672371, 2.77277190512722107479119326401, 4.39328227382920555200783205556, 6.22667110384827813339532114111, 6.59307586272633836813994618111, 7.51286732058050828218856909094, 8.827677537892191503077624826375, 9.344349154135891965786983728011, 10.30328225749213917266536410544, 11.16951035561618201740385000729
