| 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
−11.54553279696594828345961788822, −10.65502130496028422768502019018, −9.347039958475037808291303098295, −8.656673054893175220951426065222, −7.893094204533182626942445576176, −6.82804438617174762112339737786, −5.69755439852746329558928719708, −4.63213134533692845679641235062, −2.60709806475068881030596059820, −1.51676227730171882587677073167,
1.04191429777506131002487022571, 3.06659944026594203218880107763, 4.52977047922480802611809119161, 5.43823026299783194545111604532, 6.94856884650001974876851276103, 7.67829017010368357906940330598, 8.519922743747897410696341215648, 9.800901702169480819840013080855, 10.42190433009465324968779344182, 11.09147111681252212446555719277
