L(s) = 1 | + (−1.99 + 1.44i)2-s + (−0.5 − 1.53i)3-s + (1.26 − 3.88i)4-s + (2.80 + 2.03i)5-s + (3.22 + 2.34i)6-s + (−0.309 + 0.951i)7-s + (1.58 + 4.89i)8-s + (0.309 − 0.224i)9-s − 8.55·10-s + (2.91 + 1.57i)11-s − 6.60·12-s + (0.528 − 0.384i)13-s + (−0.762 − 2.34i)14-s + (1.73 − 5.33i)15-s + (−3.65 − 2.65i)16-s + (0.919 + 0.668i)17-s + ⋯ |
L(s) = 1 | + (−1.41 + 1.02i)2-s + (−0.288 − 0.888i)3-s + (0.631 − 1.94i)4-s + (1.25 + 0.911i)5-s + (1.31 + 0.957i)6-s + (−0.116 + 0.359i)7-s + (0.561 + 1.72i)8-s + (0.103 − 0.0748i)9-s − 2.70·10-s + (0.880 + 0.474i)11-s − 1.90·12-s + (0.146 − 0.106i)13-s + (−0.203 − 0.626i)14-s + (0.447 − 1.37i)15-s + (−0.913 − 0.663i)16-s + (0.223 + 0.162i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.524624 + 0.203115i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.524624 + 0.203115i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (-2.91 - 1.57i)T \) |
good | 2 | \( 1 + (1.99 - 1.44i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.5 + 1.53i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-2.80 - 2.03i)T + (1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (-0.528 + 0.384i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.919 - 0.668i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.87 + 5.77i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 6.66T + 23T^{2} \) |
| 29 | \( 1 + (1.41 - 4.34i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (2.26 - 1.64i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.135 - 0.418i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.82 + 5.61i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 + (0.186 + 0.575i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (7.94 - 5.77i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.523 + 1.61i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (5.54 + 4.03i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 6.17T + 67T^{2} \) |
| 71 | \( 1 + (-4.38 - 3.18i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.07 + 6.37i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (2.14 - 1.55i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.41 + 3.93i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 0.698T + 89T^{2} \) |
| 97 | \( 1 + (-12.0 + 8.73i)T + (29.9 - 92.2i)T^{2} \) |
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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
−14.78233938825352195553695616191, −13.86691786378767464010721161092, −12.46760864269831927070234115557, −10.88840283652130002826113649596, −9.814124713478736739395913060203, −8.999690556266380448842236767820, −7.39128558922212606943318200635, −6.58658744842989403185398835552, −5.91897077543004412185023096152, −1.83225717415857255392004335705,
1.66189493881005694540507173399, 4.01998150583811817044212011463, 5.91440232868672542697155407730, 8.050714965178102199036054866134, 9.293525355848351403247834798104, 9.810572521521794965715447585694, 10.62184632044375661839094100488, 11.82078075693188683894972515375, 12.94215819285903095537444412325, 14.19129499164814085241883577455