| L(s) = 1 | − i·2-s + 4-s − 3i·7-s − 3i·8-s + 11-s − 2i·13-s − 3·14-s − 16-s − 3i·17-s + 19-s − i·22-s + i·23-s − 2·26-s − 3i·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.5·4-s − 1.13i·7-s − 1.06i·8-s + 0.301·11-s − 0.554i·13-s − 0.801·14-s − 0.250·16-s − 0.727i·17-s + 0.229·19-s − 0.213i·22-s + 0.208i·23-s − 0.392·26-s − 0.566i·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.940183545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.940183545\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 11T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 17iT - 97T^{2} \) |
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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
−8.699408782918959626345049037874, −7.58295097358687431707525205060, −7.22270835912240985655539634680, −6.40720881365819421235420583893, −5.44057641044720505529281686741, −4.34226520463857271465691425343, −3.58166234737115796904281772330, −2.80141425953824549116473130474, −1.61714311120405321001516349843, −0.61942529144922576853363480471,
1.67935290940881962576708987127, 2.47523340911041945920181983125, 3.55221791758396881272734347657, 4.75860537385674775074185387345, 5.58248858524698529005924088828, 6.21206327067784175254319894960, 6.82065039346092720690390776456, 7.72394666564413383604960560338, 8.421432771608845102549341647750, 9.023805898248617786382142580890
