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
−10.66538265453412889701060883899, −9.509881131945936462455890581900, −8.764490341982268930205173113961, −7.83906992564692048456915742943, −6.84941215078920821747742806607, −6.04991452204686918352925728395, −5.39393663063644926434002193862, −4.54685644323301152203030844121, −3.19535472564032777864060806824, −2.05687836546802658829696868505,
0.18609990693619401354458034938, 2.14538900617180450261774025339, 3.25556472399695291078808683090, 4.33622185468746807200802600848, 5.07719626646807567230029580135, 6.29300324270489074828726946443, 6.73319900602033727586315995629, 7.81599422331129653933702305328, 9.108668599882498467360159736318, 10.07904905247150392910340649019