| L(s) = 1 | + (−0.978 + 0.207i)2-s + (−0.309 + 0.951i)3-s + (0.913 − 0.406i)4-s + (−0.0507 + 0.482i)5-s + (0.104 − 0.994i)6-s + (3.08 − 3.42i)7-s + (−0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.0507 − 0.482i)10-s + 0.474·11-s + (0.104 + 0.994i)12-s + (−2.23 − 3.87i)13-s + (−2.30 + 3.99i)14-s + (−0.443 − 0.197i)15-s + (0.669 − 0.743i)16-s + (3.16 − 1.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 + 0.147i)2-s + (−0.178 + 0.549i)3-s + (0.456 − 0.203i)4-s + (−0.0226 + 0.215i)5-s + (0.0426 − 0.406i)6-s + (1.16 − 1.29i)7-s + (−0.286 + 0.207i)8-s + (−0.269 − 0.195i)9-s + (−0.0160 − 0.152i)10-s + 0.142·11-s + (0.0301 + 0.287i)12-s + (−0.620 − 1.07i)13-s + (−0.616 + 1.06i)14-s + (−0.114 − 0.0509i)15-s + (0.167 − 0.185i)16-s + (0.768 − 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02019 - 0.0789430i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02019 - 0.0789430i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 - 0.207i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 61 | \( 1 + (-7.75 - 0.942i)T \) |
| good | 5 | \( 1 + (0.0507 - 0.482i)T + (-4.89 - 1.03i)T^{2} \) |
| 7 | \( 1 + (-3.08 + 3.42i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 - 0.474T + 11T^{2} \) |
| 13 | \( 1 + (2.23 + 3.87i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.16 + 1.41i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (0.713 + 0.791i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-3.08 - 2.23i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.725 + 1.25i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.47 - 0.525i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (-1.93 - 5.95i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.412 + 1.26i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (0.221 + 0.0984i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (-0.151 + 0.262i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.44 + 2.50i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (12.6 - 2.69i)T + (53.8 - 23.9i)T^{2} \) |
| 67 | \( 1 + (0.263 - 2.50i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-0.698 - 6.64i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (0.336 + 3.20i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (12.9 + 5.74i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (13.1 - 2.78i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (0.741 - 2.28i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-10.1 - 2.14i)T + (88.6 + 39.4i)T^{2} \) |
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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
−11.09147111681252212446555719277, −10.42190433009465324968779344182, −9.800901702169480819840013080855, −8.519922743747897410696341215648, −7.67829017010368357906940330598, −6.94856884650001974876851276103, −5.43823026299783194545111604532, −4.52977047922480802611809119161, −3.06659944026594203218880107763, −1.04191429777506131002487022571,
1.51676227730171882587677073167, 2.60709806475068881030596059820, 4.63213134533692845679641235062, 5.69755439852746329558928719708, 6.82804438617174762112339737786, 7.893094204533182626942445576176, 8.656673054893175220951426065222, 9.347039958475037808291303098295, 10.65502130496028422768502019018, 11.54553279696594828345961788822
