L(s) = 1 | + (−1.93 − 1.40i)2-s + (1.15 + 3.55i)4-s + (−0.809 + 0.587i)5-s + (0.608 + 1.87i)7-s + (1.28 − 3.95i)8-s + 2.39·10-s + (1.69 − 2.84i)11-s + (−2.36 − 1.71i)13-s + (1.45 − 4.48i)14-s + (−2.00 + 1.45i)16-s + (−5.35 + 3.89i)17-s + (−1.10 + 3.38i)19-s + (−3.02 − 2.19i)20-s + (−7.30 + 3.13i)22-s − 2.78·23-s + ⋯ |
L(s) = 1 | + (−1.37 − 0.995i)2-s + (0.577 + 1.77i)4-s + (−0.361 + 0.262i)5-s + (0.230 + 0.708i)7-s + (0.454 − 1.39i)8-s + 0.757·10-s + (0.511 − 0.859i)11-s + (−0.655 − 0.476i)13-s + (0.389 − 1.19i)14-s + (−0.502 + 0.364i)16-s + (−1.29 + 0.943i)17-s + (−0.252 + 0.776i)19-s + (−0.675 − 0.490i)20-s + (−1.55 + 0.667i)22-s − 0.581·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.149069 + 0.167232i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.149069 + 0.167232i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 + (-1.69 + 2.84i)T \) |
good | 2 | \( 1 + (1.93 + 1.40i)T + (0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 + (-0.608 - 1.87i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (2.36 + 1.71i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (5.35 - 3.89i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.10 - 3.38i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 + (-1.08 - 3.34i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (6.56 + 4.77i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.36 - 4.21i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.18 - 6.73i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + (-0.00839 + 0.0258i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.32 - 2.41i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.94 - 9.04i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (2.50 - 1.82i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 1.58T + 67T^{2} \) |
| 71 | \( 1 + (4.90 - 3.56i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.733 + 2.25i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-10.1 - 7.37i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (4.08 - 2.96i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + (8.42 + 6.11i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.07750324202393072083502354023, −10.34255476414923208334627229521, −9.457823353458133081272092543699, −8.472599984307006338710924116031, −8.190078035188119592565224128848, −6.90265423600630726541933503339, −5.70407909617648610270588234654, −4.00597701819109917378777581848, −2.85535301101758411832376947373, −1.71147405060159010456523058955,
0.20481243555999796316986383133, 1.93439612979499160636113882985, 4.16827962659178074221562212320, 5.14922874424323105140981454199, 6.80306868606327730755571079377, 6.99362779519268519929425461453, 7.952002772367330943866765463076, 8.973426019384813239566919747235, 9.468757419785183797716738115983, 10.42824811123230105008073281700