| L(s) = 1 | + (0.755 + 1.85i)2-s + (0.704 − 2.91i)3-s + (−2.85 + 2.79i)4-s + (1.59 + 9.02i)5-s + (5.93 − 0.900i)6-s + (−0.928 − 0.778i)7-s + (−7.34 − 3.17i)8-s + (−8.00 − 4.10i)9-s + (−15.5 + 9.77i)10-s + (−3.17 + 17.9i)11-s + (6.15 + 10.3i)12-s + (4.20 − 11.5i)13-s + (0.740 − 2.30i)14-s + (27.4 + 1.71i)15-s + (0.330 − 15.9i)16-s + (−16.5 + 9.56i)17-s + ⋯ |
| L(s) = 1 | + (0.377 + 0.925i)2-s + (0.234 − 0.972i)3-s + (−0.714 + 0.699i)4-s + (0.318 + 1.80i)5-s + (0.988 − 0.150i)6-s + (−0.132 − 0.111i)7-s + (−0.917 − 0.396i)8-s + (−0.889 − 0.456i)9-s + (−1.55 + 0.977i)10-s + (−0.288 + 1.63i)11-s + (0.512 + 0.858i)12-s + (0.323 − 0.889i)13-s + (0.0529 − 0.164i)14-s + (1.82 + 0.114i)15-s + (0.0206 − 0.999i)16-s + (−0.974 + 0.562i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.650934 + 1.48155i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.650934 + 1.48155i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.755 - 1.85i)T \) |
| 3 | \( 1 + (-0.704 + 2.91i)T \) |
| good | 5 | \( 1 + (-1.59 - 9.02i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (0.928 + 0.778i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (3.17 - 17.9i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-4.20 + 11.5i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (16.5 - 9.56i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-21.0 - 12.1i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-15.5 - 18.5i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-3.71 + 1.35i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-17.2 + 14.4i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (5.34 - 3.08i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-10.7 + 29.6i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-55.2 - 9.74i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-10.6 + 12.7i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 55.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (5.56 + 31.5i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (13.9 - 16.6i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-11.1 + 30.6i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-62.2 + 35.9i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (19.3 - 33.4i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-65.6 + 23.9i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (25.2 - 9.17i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (104. + 60.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (31.1 - 176. i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−12.77751964960679946113259960755, −11.71065812854540866164161070228, −10.45230210075039401987280385626, −9.419482090149918534308346312178, −7.86706800995853105382144248666, −7.26119369674237861656338095587, −6.56574861912995552759639705075, −5.56195363054861114520056940822, −3.61019894450089673816351161309, −2.43558627779728927911265677476,
0.793180145258287749150965723240, 2.78316953719374584882471183011, 4.24301145419967576585844921437, 5.01382926939687019606993281534, 5.91458736396314103501574861554, 8.556998112564049786691998420051, 8.922650462454195441337294231291, 9.624410282758861186072216717442, 10.96016801317744463530070452774, 11.58611350626223425492699053757
