| 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
−15.29434869846482026936061467446, −14.17228653362424909598390256221, −12.81996106690395657179456588826, −12.01535786136574311546998145840, −10.38672711815453548691190923752, −9.312245010562683487348875121216, −8.112936293453308943571023188345, −7.17780956813609651259461951870, −4.40181658131456009225218614586, −2.51670461556130474050529429233,
3.06871532710747690168428744880, 5.24989359102072482881368984818, 7.59186670662873058684024851683, 7.87287577341603749603827321112, 9.573098189475463649810978116765, 10.39402731653935810294583815558, 12.09089867386352871031740285650, 13.56130076475019902938084495035, 14.59160954863557728948433972895, 15.55984201715101010373257725906
