L(s) = 1 | + (1.87 + 0.684i)4-s + (3.75 − 1.36i)7-s + (−1.21 − 6.89i)13-s + (3.06 + 2.57i)16-s + (0.5 − 0.866i)19-s + (−0.868 + 4.92i)25-s + 8·28-s + (−10.3 − 3.76i)31-s + (5 + 8.66i)37-s + (3.83 + 3.21i)43-s + (6.89 − 5.78i)49-s + (2.43 − 13.7i)52-s + (0.939 − 0.342i)61-s + (4.00 + 6.92i)64-s + (0.868 + 4.92i)67-s + ⋯ |
L(s) = 1 | + (0.939 + 0.342i)4-s + (1.42 − 0.517i)7-s + (−0.337 − 1.91i)13-s + (0.766 + 0.642i)16-s + (0.114 − 0.198i)19-s + (−0.173 + 0.984i)25-s + 1.51·28-s + (−1.85 − 0.675i)31-s + (0.821 + 1.42i)37-s + (0.584 + 0.490i)43-s + (0.984 − 0.826i)49-s + (0.337 − 1.91i)52-s + (0.120 − 0.0437i)61-s + (0.500 + 0.866i)64-s + (0.106 + 0.601i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.13599 - 0.249661i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.13599 - 0.249661i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-3.75 + 1.36i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.21 + 6.89i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (10.3 + 3.76i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.83 - 3.21i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.939 + 0.342i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.868 - 4.92i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.25 - 12.8i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.83 - 3.21i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.70925226193231354456272021353, −9.634733782855660582524628932463, −8.251524337115431433441680777913, −7.77599912709894921442483145417, −7.14853762790095540475021403753, −5.81323187063364578738272510163, −5.04902550145165744703040859174, −3.73357354768251862478401609262, −2.61517623242230174196041248963, −1.30855079418430445403231739178,
1.67746716554469764605378563990, 2.31871220956846850570265009291, 4.05019558106758557138363098820, 5.08023397356834089827238608801, 5.97779230704522323554785529621, 7.00043667889031147793486446017, 7.69012995415580820464636569990, 8.754391096303050862800700849369, 9.509916607610505459919290015648, 10.69916940274756016267182009297