| L(s) = 1 | + (−0.892 + 1.09i)2-s + (−0.406 − 1.95i)4-s − 2.56i·5-s + (2.51 + 1.30i)8-s + (2.81 + 2.28i)10-s − 1.15·11-s + 0.578·13-s + (−3.66 + 1.59i)16-s − 5.39i·17-s − 6.20i·19-s + (−5.02 + 1.04i)20-s + (1.02 − 1.26i)22-s − 7.62·23-s − 1.57·25-s + (−0.516 + 0.634i)26-s + ⋯ |
| L(s) = 1 | + (−0.631 + 0.775i)2-s + (−0.203 − 0.979i)4-s − 1.14i·5-s + (0.887 + 0.460i)8-s + (0.889 + 0.723i)10-s − 0.346·11-s + 0.160·13-s + (−0.917 + 0.398i)16-s − 1.30i·17-s − 1.42i·19-s + (−1.12 + 0.233i)20-s + (0.218 − 0.269i)22-s − 1.58·23-s − 0.315·25-s + (−0.101 + 0.124i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5293949689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5293949689\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.892 - 1.09i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2.56iT - 5T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 - 0.578T + 13T^{2} \) |
| 17 | \( 1 + 5.39iT - 17T^{2} \) |
| 19 | \( 1 + 6.20iT - 19T^{2} \) |
| 23 | \( 1 + 7.62T + 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 5.04iT - 31T^{2} \) |
| 37 | \( 1 - 9.83T + 37T^{2} \) |
| 41 | \( 1 - 6.21iT - 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 4.53iT - 53T^{2} \) |
| 59 | \( 1 + 4.83T + 59T^{2} \) |
| 61 | \( 1 + 0.951T + 61T^{2} \) |
| 67 | \( 1 + 2.78iT - 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 12.8iT - 79T^{2} \) |
| 83 | \( 1 + 8.77T + 83T^{2} \) |
| 89 | \( 1 + 5.68iT - 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−9.017339087336681251027894444559, −8.166623453543630820109529516311, −7.65115202135809804024380579878, −6.63647018593665375134352352908, −5.87610790640456029652275854274, −4.78845077091806097518590268706, −4.61684220633556982084554754163, −2.79548133572361516618870394425, −1.39778406206600076957329166001, −0.25084258628860063816319786474,
1.66820493613374074843700494452, 2.54385274336286086444442798381, 3.61581560067287319567334451247, 4.15021645573010844626920695474, 5.75513537909771646734648368398, 6.44939086755908693117044016202, 7.54096486571946382027698929325, 8.003620707219400106690860889905, 8.810353311624131083032537840775, 10.01929325416176223497179879087
