| L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.913 + 0.406i)3-s + (0.309 − 0.951i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (−1.80 − 2.00i)7-s + (−0.309 − 0.951i)8-s + (0.669 − 0.743i)9-s + (0.913 + 0.406i)10-s + (−3.30 + 0.703i)11-s + (0.104 + 0.994i)12-s + (−0.0272 + 0.259i)13-s + (−2.63 − 0.560i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (−6.17 − 1.31i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.527 + 0.234i)3-s + (0.154 − 0.475i)4-s + (0.223 + 0.387i)5-s + (−0.204 + 0.353i)6-s + (−0.681 − 0.757i)7-s + (−0.109 − 0.336i)8-s + (0.223 − 0.247i)9-s + (0.288 + 0.128i)10-s + (−0.997 + 0.212i)11-s + (0.0301 + 0.287i)12-s + (−0.00755 + 0.0718i)13-s + (−0.704 − 0.149i)14-s + (−0.208 − 0.151i)15-s + (−0.202 − 0.146i)16-s + (−1.49 − 0.318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0197772 - 0.493995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0197772 - 0.493995i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.913 - 0.406i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.423 + 5.55i)T \) |
| good | 7 | \( 1 + (1.80 + 2.00i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (3.30 - 0.703i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.0272 - 0.259i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (6.17 + 1.31i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.114 - 1.08i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (1.53 + 4.72i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (0.0165 - 0.0120i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (5.17 - 8.95i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.831 - 0.370i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.399 + 3.79i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (5.27 + 3.83i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (0.960 - 1.06i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (5.48 - 2.44i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + (2.25 + 3.90i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.08 + 4.53i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-13.4 + 2.85i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-13.3 - 2.83i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (12.4 + 5.54i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (2.76 - 8.49i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.95 + 15.2i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.07904905247150392910340649019, −9.108668599882498467360159736318, −7.81599422331129653933702305328, −6.73319900602033727586315995629, −6.29300324270489074828726946443, −5.07719626646807567230029580135, −4.33622185468746807200802600848, −3.25556472399695291078808683090, −2.14538900617180450261774025339, −0.18609990693619401354458034938,
2.05687836546802658829696868505, 3.19535472564032777864060806824, 4.54685644323301152203030844121, 5.39393663063644926434002193862, 6.04991452204686918352925728395, 6.84941215078920821747742806607, 7.83906992564692048456915742943, 8.764490341982268930205173113961, 9.509881131945936462455890581900, 10.66538265453412889701060883899
