| L(s) = 1 | + (0.256 + 1.39i)2-s + (1.72 − 0.182i)3-s + (−1.86 + 0.713i)4-s + (0.370 + 2.81i)5-s + (0.695 + 2.34i)6-s + (0.178 + 0.665i)7-s + (−1.47 − 2.41i)8-s + (2.93 − 0.627i)9-s + (−3.81 + 1.23i)10-s + (−2.18 + 1.67i)11-s + (−3.08 + 1.56i)12-s + (3.45 − 4.50i)13-s + (−0.879 + 0.418i)14-s + (1.15 + 4.77i)15-s + (2.98 − 2.66i)16-s − 6.90·17-s + ⋯ |
| L(s) = 1 | + (0.181 + 0.983i)2-s + (0.994 − 0.105i)3-s + (−0.934 + 0.356i)4-s + (0.165 + 1.25i)5-s + (0.283 + 0.958i)6-s + (0.0673 + 0.251i)7-s + (−0.520 − 0.854i)8-s + (0.977 − 0.209i)9-s + (−1.20 + 0.391i)10-s + (−0.657 + 0.504i)11-s + (−0.891 + 0.453i)12-s + (0.958 − 1.24i)13-s + (−0.235 + 0.111i)14-s + (0.297 + 1.23i)15-s + (0.745 − 0.666i)16-s − 1.67·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.997559 + 1.43994i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.997559 + 1.43994i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.256 - 1.39i)T \) |
| 3 | \( 1 + (-1.72 + 0.182i)T \) |
| good | 5 | \( 1 + (-0.370 - 2.81i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (-0.178 - 0.665i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (2.18 - 1.67i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-3.45 + 4.50i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 + 6.90T + 17T^{2} \) |
| 19 | \( 1 + (2.38 - 0.987i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-6.36 - 1.70i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-9.39 - 1.23i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (3.44 - 1.98i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.725 + 1.75i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-1.12 + 4.20i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (6.42 + 8.37i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (6.13 + 3.54i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.365 + 0.881i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (0.242 - 0.0319i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-0.772 + 5.87i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-2.17 + 2.83i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-1.97 - 1.97i)T + 71iT^{2} \) |
| 73 | \( 1 + (-0.741 + 0.741i)T - 73iT^{2} \) |
| 79 | \( 1 + (4.86 - 8.42i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.345 - 0.0454i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (2.50 - 2.50i)T - 89iT^{2} \) |
| 97 | \( 1 + (2.87 - 4.98i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.59407197180865623677641862646, −10.84266384556109880994778462946, −10.16108332441042712667620750940, −8.863763052881243665974774000282, −8.248711545085452060921659452186, −7.09409053542256972965112596073, −6.55630663462298018838456778040, −5.10769110625798258493044619078, −3.65041565226374704014754744873, −2.60780150821703371433793671753,
1.36072032942443408037965422834, 2.73981438927689640221081733730, 4.28287867053529767544363468052, 4.76772271662178638139552272024, 6.51663404799766590145695059022, 8.343726744763972325742333973964, 8.763540255849896864802870398824, 9.442105470577671091006234618455, 10.66882278048625914518093156908, 11.43533772175867049279110438572
