| L(s) = 1 | + (−1.75 − 1.88i)2-s + (0.0116 + 1.73i)3-s + (−0.346 + 4.61i)4-s + (−0.695 − 0.871i)5-s + (3.24 − 3.05i)6-s + (−2.60 + 0.480i)7-s + (5.29 − 4.22i)8-s + (−2.99 + 0.0402i)9-s + (−0.428 + 2.83i)10-s + (1.68 − 0.383i)11-s + (−8.00 − 0.545i)12-s + (2.33 + 2.52i)13-s + (5.46 + 4.07i)14-s + (1.50 − 1.21i)15-s + (−8.09 − 1.21i)16-s + (−0.251 − 3.34i)17-s + ⋯ |
| L(s) = 1 | + (−1.23 − 1.33i)2-s + (0.00671 + 0.999i)3-s + (−0.173 + 2.30i)4-s + (−0.310 − 0.389i)5-s + (1.32 − 1.24i)6-s + (−0.983 + 0.181i)7-s + (1.87 − 1.49i)8-s + (−0.999 + 0.0134i)9-s + (−0.135 + 0.898i)10-s + (0.507 − 0.115i)11-s + (−2.31 − 0.157i)12-s + (0.648 + 0.699i)13-s + (1.46 + 1.08i)14-s + (0.387 − 0.313i)15-s + (−2.02 − 0.304i)16-s + (−0.0608 − 0.812i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0668239 - 0.278527i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0668239 - 0.278527i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.0116 - 1.73i)T \) |
| 7 | \( 1 + (2.60 - 0.480i)T \) |
| good | 2 | \( 1 + (1.75 + 1.88i)T + (-0.149 + 1.99i)T^{2} \) |
| 5 | \( 1 + (0.695 + 0.871i)T + (-1.11 + 4.87i)T^{2} \) |
| 11 | \( 1 + (-1.68 + 0.383i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-2.33 - 2.52i)T + (-0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (0.251 + 3.34i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (2.18 + 1.26i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.848 - 1.76i)T + (-14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (4.77 + 7.00i)T + (-10.5 + 26.9i)T^{2} \) |
| 31 | \( 1 + (9.00 + 5.20i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.28 + 1.55i)T + (13.5 - 34.4i)T^{2} \) |
| 41 | \( 1 + (-2.19 + 5.59i)T + (-30.0 - 27.8i)T^{2} \) |
| 43 | \( 1 + (-0.249 - 0.636i)T + (-31.5 + 29.2i)T^{2} \) |
| 47 | \( 1 + (-4.14 + 3.84i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (6.46 - 9.48i)T + (-19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (2.63 + 6.71i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-3.05 + 0.229i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (4.30 - 7.45i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.09 + 6.42i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (3.45 + 11.2i)T + (-60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (-6.65 - 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.945 - 0.876i)T + (6.20 + 82.7i)T^{2} \) |
| 89 | \( 1 + (10.3 + 9.58i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + (-11.4 - 6.60i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.73853735623699834516497857916, −9.636562499859293894152238761108, −9.247116600581487981694690586889, −8.669033311882899208022213780674, −7.46275379998977436912508166937, −6.00033578665892560987015096289, −4.24704198103396789918237947440, −3.57695726599496061203996607195, −2.34523218572643386261203485267, −0.28487704784686768184293728888,
1.39260916804823898888362396352, 3.42123684088985320338219888852, 5.55595626510483271643412205983, 6.33606889824228520215033920083, 6.98525749467682769973802166346, 7.68955792628854656100611605863, 8.647429629923249644488381062188, 9.262565111127489827054776572544, 10.49671372978504970628654229440, 11.08151461814515250404433236066
