| L(s) = 1 | + (−0.766 + 0.642i)2-s + (1.70 + 0.300i)3-s + (0.173 − 0.984i)4-s + (−1.26 + 0.460i)5-s + (−1.5 + 0.866i)6-s + (−0.0209 − 0.118i)7-s + (0.500 + 0.866i)8-s + (2.81 + 1.02i)9-s + (0.673 − 1.16i)10-s + (−3.49 − 1.27i)11-s + (0.592 − 1.62i)12-s + (−4.64 − 3.89i)13-s + (0.0923 + 0.0775i)14-s + (−2.29 + 0.405i)15-s + (−0.939 − 0.342i)16-s + (2.58 − 4.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.984 + 0.173i)3-s + (0.0868 − 0.492i)4-s + (−0.566 + 0.206i)5-s + (−0.612 + 0.353i)6-s + (−0.00791 − 0.0448i)7-s + (0.176 + 0.306i)8-s + (0.939 + 0.342i)9-s + (0.213 − 0.368i)10-s + (−1.05 − 0.383i)11-s + (0.171 − 0.469i)12-s + (−1.28 − 1.08i)13-s + (0.0246 + 0.0207i)14-s + (−0.593 + 0.104i)15-s + (−0.234 − 0.0855i)16-s + (0.626 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.760434 + 0.227659i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.760434 + 0.227659i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (-1.70 - 0.300i)T \) |
| good | 5 | \( 1 + (1.26 - 0.460i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.0209 + 0.118i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (3.49 + 1.27i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (4.64 + 3.89i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.58 + 4.47i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.96 - 5.12i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.826 - 4.68i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.55 + 3.82i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.875 - 4.96i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.145 - 0.251i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.44 - 3.72i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.426 + 0.155i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.134 + 0.761i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 7.29T + 53T^{2} \) |
| 59 | \( 1 + (-1.40 + 0.509i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.656 + 3.72i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (5.08 + 4.26i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.87 - 4.97i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.20 + 9.02i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.7 + 8.99i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (1.81 - 1.52i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.08 - 1.87i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.21 - 1.17i)T + (74.3 + 62.3i)T^{2} \) |
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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
−15.55984201715101010373257725906, −14.59160954863557728948433972895, −13.56130076475019902938084495035, −12.09089867386352871031740285650, −10.39402731653935810294583815558, −9.573098189475463649810978116765, −7.87287577341603749603827321112, −7.59186670662873058684024851683, −5.24989359102072482881368984818, −3.06871532710747690168428744880,
2.51670461556130474050529429233, 4.40181658131456009225218614586, 7.17780956813609651259461951870, 8.112936293453308943571023188345, 9.312245010562683487348875121216, 10.38672711815453548691190923752, 12.01535786136574311546998145840, 12.81996106690395657179456588826, 14.17228653362424909598390256221, 15.29434869846482026936061467446
