| L(s) = 1 | + (1.72 − 0.162i)3-s + 5-s + 4.26i·7-s + (2.94 − 0.561i)9-s + 2.35·11-s − 0.194i·13-s + (1.72 − 0.162i)15-s + 0.326i·17-s + 7.96·19-s + (0.695 + 7.35i)21-s + 6.28i·23-s + 25-s + (4.99 − 1.44i)27-s − 3.08i·29-s − 1.32i·31-s + ⋯ |
| L(s) = 1 | + (0.995 − 0.0940i)3-s + 0.447·5-s + 1.61i·7-s + (0.982 − 0.187i)9-s + 0.710·11-s − 0.0539i·13-s + (0.445 − 0.0420i)15-s + 0.0792i·17-s + 1.82·19-s + (0.151 + 1.60i)21-s + 1.30i·23-s + 0.200·25-s + (0.960 − 0.278i)27-s − 0.573i·29-s − 0.238i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.471986896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.471986896\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.72 + 0.162i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (6.00 - 5.56i)T \) |
| good | 7 | \( 1 - 4.26iT - 7T^{2} \) |
| 11 | \( 1 - 2.35T + 11T^{2} \) |
| 13 | \( 1 + 0.194iT - 13T^{2} \) |
| 17 | \( 1 - 0.326iT - 17T^{2} \) |
| 19 | \( 1 - 7.96T + 19T^{2} \) |
| 23 | \( 1 - 6.28iT - 23T^{2} \) |
| 29 | \( 1 + 3.08iT - 29T^{2} \) |
| 31 | \( 1 + 1.32iT - 31T^{2} \) |
| 37 | \( 1 + 3.97T + 37T^{2} \) |
| 41 | \( 1 + 7.68T + 41T^{2} \) |
| 43 | \( 1 + 12.0iT - 43T^{2} \) |
| 47 | \( 1 - 10.6iT - 47T^{2} \) |
| 53 | \( 1 + 7.78T + 53T^{2} \) |
| 59 | \( 1 + 9.12iT - 59T^{2} \) |
| 61 | \( 1 - 8.50iT - 61T^{2} \) |
| 71 | \( 1 + 5.38iT - 71T^{2} \) |
| 73 | \( 1 - 9.17T + 73T^{2} \) |
| 79 | \( 1 + 9.07iT - 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 + 6.98iT - 89T^{2} \) |
| 97 | \( 1 - 4.00iT - 97T^{2} \) |
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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
−8.677523868659984399772133442596, −7.895693426668909751211145585930, −7.21324449843733392014393809436, −6.30662125476612210974321867647, −5.56566272705305221332771492535, −4.91594950046447145734118142315, −3.63139585758135052761966934876, −3.06676040588163578014841812096, −2.12685557052728599588290116211, −1.39409352643282830747547038489,
0.950871293428180321823919444984, 1.72591478914433119573825117080, 3.06070448540193776666425599918, 3.57691909647966708853460582183, 4.47326243054786778677368682640, 5.10617102885283755730458762919, 6.47217544997227074329547644087, 6.96803312059298646757328414582, 7.59543018746121477890965010570, 8.331239683320939371705677400448
