| L(s) = 1 | + (−2.38 − 0.0307i)2-s + (−1.09 + 1.33i)3-s + (3.66 + 0.0947i)4-s + (2.99 + 0.916i)5-s + (2.65 − 3.15i)6-s + (−0.120 + 0.301i)7-s + (−3.96 − 0.153i)8-s + (−0.580 − 2.94i)9-s + (−7.09 − 2.27i)10-s + (−2.57 − 1.46i)11-s + (−4.15 + 4.79i)12-s + (2.35 + 5.26i)13-s + (0.297 − 0.714i)14-s + (−4.51 + 2.99i)15-s + (2.10 + 0.108i)16-s + (−6.15 − 2.79i)17-s + ⋯ |
| L(s) = 1 | + (−1.68 − 0.0217i)2-s + (−0.635 + 0.772i)3-s + (1.83 + 0.0473i)4-s + (1.33 + 0.410i)5-s + (1.08 − 1.28i)6-s + (−0.0457 + 0.114i)7-s + (−1.40 − 0.0543i)8-s + (−0.193 − 0.981i)9-s + (−2.24 − 0.719i)10-s + (−0.775 − 0.441i)11-s + (−1.20 + 1.38i)12-s + (0.651 + 1.45i)13-s + (0.0794 − 0.190i)14-s + (−1.16 + 0.773i)15-s + (0.525 + 0.0272i)16-s + (−1.49 − 0.678i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0499886 + 0.354523i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0499886 + 0.354523i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.09 - 1.33i)T \) |
| good | 2 | \( 1 + (2.38 + 0.0307i)T + (1.99 + 0.0517i)T^{2} \) |
| 5 | \( 1 + (-2.99 - 0.916i)T + (4.14 + 2.80i)T^{2} \) |
| 7 | \( 1 + (0.120 - 0.301i)T + (-5.06 - 4.83i)T^{2} \) |
| 11 | \( 1 + (2.57 + 1.46i)T + (5.62 + 9.45i)T^{2} \) |
| 13 | \( 1 + (-2.35 - 5.26i)T + (-8.67 + 9.68i)T^{2} \) |
| 17 | \( 1 + (6.15 + 2.79i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.549 + 0.801i)T + (-6.84 + 17.7i)T^{2} \) |
| 23 | \( 1 + (1.84 - 5.61i)T + (-18.5 - 13.6i)T^{2} \) |
| 29 | \( 1 + (3.83 - 2.45i)T + (12.1 - 26.3i)T^{2} \) |
| 31 | \( 1 + (0.244 + 0.901i)T + (-26.7 + 15.6i)T^{2} \) |
| 37 | \( 1 + (-10.3 - 2.03i)T + (34.2 + 13.9i)T^{2} \) |
| 41 | \( 1 + (0.489 - 0.185i)T + (30.7 - 27.1i)T^{2} \) |
| 43 | \( 1 + (8.37 - 3.67i)T + (29.1 - 31.6i)T^{2} \) |
| 47 | \( 1 + (-2.08 + 7.66i)T + (-40.5 - 23.7i)T^{2} \) |
| 53 | \( 1 + (8.14 - 8.62i)T + (-3.08 - 52.9i)T^{2} \) |
| 59 | \( 1 + (9.23 - 1.56i)T + (55.6 - 19.4i)T^{2} \) |
| 61 | \( 1 + (5.54 - 10.1i)T + (-33.1 - 51.1i)T^{2} \) |
| 67 | \( 1 + (10.7 - 5.55i)T + (38.6 - 54.7i)T^{2} \) |
| 71 | \( 1 + (-1.85 - 1.43i)T + (17.7 + 68.7i)T^{2} \) |
| 73 | \( 1 + (-6.09 + 1.95i)T + (59.3 - 42.4i)T^{2} \) |
| 79 | \( 1 + (0.343 - 1.81i)T + (-73.5 - 28.9i)T^{2} \) |
| 83 | \( 1 + (-0.252 - 1.54i)T + (-78.7 + 26.3i)T^{2} \) |
| 89 | \( 1 + (0.100 + 0.0411i)T + (63.5 + 62.3i)T^{2} \) |
| 97 | \( 1 + (-2.87 + 12.5i)T + (-87.2 - 42.4i)T^{2} \) |
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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
−10.65326377695943236525353475693, −9.715185342373643663377980034861, −9.300289878739584725189374174460, −8.709871509812717893031340864011, −7.31803807597091699881177890269, −6.42126191565395931005930267490, −5.85455883390802520991191632718, −4.50526065584535509609769472218, −2.76343339312868906205806898723, −1.62077676973608588947824951739,
0.33348197255408181125529279018, 1.70398962087264485150287968899, 2.44050141939483578298956349374, 4.88983758576312545068128368315, 6.06729459263315421940879467949, 6.44014395973310459718507425432, 7.71810067204262270307483706930, 8.234152206146817727832920972582, 9.135304512308785046558262691520, 10.08471492457491853893110018888
