| L(s) = 1 | + (−1.90 − 0.0246i)2-s + (0.0146 − 1.73i)3-s + (1.64 + 0.0425i)4-s + (−2.27 − 0.696i)5-s + (−0.0706 + 3.30i)6-s + (0.132 − 0.330i)7-s + (0.677 + 0.0262i)8-s + (−2.99 − 0.0505i)9-s + (4.31 + 1.38i)10-s + (−4.51 − 2.56i)11-s + (0.0976 − 2.84i)12-s + (1.99 + 4.46i)13-s + (−0.261 + 0.627i)14-s + (−1.23 + 3.92i)15-s + (−4.57 − 0.236i)16-s + (0.761 + 0.346i)17-s + ⋯ |
| L(s) = 1 | + (−1.34 − 0.0174i)2-s + (0.00843 − 0.999i)3-s + (0.822 + 0.0212i)4-s + (−1.01 − 0.311i)5-s + (−0.0288 + 1.34i)6-s + (0.0501 − 0.124i)7-s + (0.239 + 0.00929i)8-s + (−0.999 − 0.0168i)9-s + (1.36 + 0.437i)10-s + (−1.35 − 0.773i)11-s + (0.0281 − 0.821i)12-s + (0.553 + 1.23i)13-s + (−0.0698 + 0.167i)14-s + (−0.319 + 1.01i)15-s + (−1.14 − 0.0592i)16-s + (0.184 + 0.0839i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.240597 + 0.0818951i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.240597 + 0.0818951i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.0146 + 1.73i)T \) |
| good | 2 | \( 1 + (1.90 + 0.0246i)T + (1.99 + 0.0517i)T^{2} \) |
| 5 | \( 1 + (2.27 + 0.696i)T + (4.14 + 2.80i)T^{2} \) |
| 7 | \( 1 + (-0.132 + 0.330i)T + (-5.06 - 4.83i)T^{2} \) |
| 11 | \( 1 + (4.51 + 2.56i)T + (5.62 + 9.45i)T^{2} \) |
| 13 | \( 1 + (-1.99 - 4.46i)T + (-8.67 + 9.68i)T^{2} \) |
| 17 | \( 1 + (-0.761 - 0.346i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.376 + 0.548i)T + (-6.84 + 17.7i)T^{2} \) |
| 23 | \( 1 + (-0.0415 + 0.126i)T + (-18.5 - 13.6i)T^{2} \) |
| 29 | \( 1 + (4.04 - 2.58i)T + (12.1 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-1.19 - 4.41i)T + (-26.7 + 15.6i)T^{2} \) |
| 37 | \( 1 + (5.74 + 1.12i)T + (34.2 + 13.9i)T^{2} \) |
| 41 | \( 1 + (-8.11 + 3.07i)T + (30.7 - 27.1i)T^{2} \) |
| 43 | \( 1 + (-6.88 + 3.02i)T + (29.1 - 31.6i)T^{2} \) |
| 47 | \( 1 + (-0.700 + 2.58i)T + (-40.5 - 23.7i)T^{2} \) |
| 53 | \( 1 + (8.09 - 8.57i)T + (-3.08 - 52.9i)T^{2} \) |
| 59 | \( 1 + (-2.21 + 0.376i)T + (55.6 - 19.4i)T^{2} \) |
| 61 | \( 1 + (-2.04 + 3.76i)T + (-33.1 - 51.1i)T^{2} \) |
| 67 | \( 1 + (-6.15 + 3.18i)T + (38.6 - 54.7i)T^{2} \) |
| 71 | \( 1 + (-1.24 - 0.963i)T + (17.7 + 68.7i)T^{2} \) |
| 73 | \( 1 + (-8.81 + 2.82i)T + (59.3 - 42.4i)T^{2} \) |
| 79 | \( 1 + (1.29 - 6.85i)T + (-73.5 - 28.9i)T^{2} \) |
| 83 | \( 1 + (-0.886 - 5.43i)T + (-78.7 + 26.3i)T^{2} \) |
| 89 | \( 1 + (-10.8 - 4.45i)T + (63.5 + 62.3i)T^{2} \) |
| 97 | \( 1 + (1.91 - 8.32i)T + (-87.2 - 42.4i)T^{2} \) |
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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
−10.77983808613708230734025820318, −9.225850903799154911672094914208, −8.677381112946676362147817716865, −7.898231240994410743857285828080, −7.47584643194852582226596958426, −6.49350794755002404583578334067, −5.22104647976550335193221056227, −3.83270554930575293421986639845, −2.32395869300091926062554334084, −0.959671063674835446827853174982,
0.26640029058534003255054411272, 2.51041816456173296671106754295, 3.71257071955863388909837669086, 4.77756604959760741023072724117, 5.81989438668062617136422281243, 7.41319959099094808267452857553, 7.909695678781118758289590166007, 8.516250931644631905811739436993, 9.623005741864555255452362756767, 10.15165028607224997999549087108
