L(s) = 1 | + (−0.425 + 0.737i)2-s + (−0.330 − 0.572i)3-s + (0.637 + 1.10i)4-s + (1.72 − 2.98i)5-s + 0.562·6-s + (−0.751 − 2.53i)7-s − 2.78·8-s + (1.28 − 2.21i)9-s + (1.46 + 2.53i)10-s + (−0.448 − 0.777i)11-s + (0.421 − 0.730i)12-s + (2.18 + 0.525i)14-s − 2.27·15-s + (−0.0891 + 0.154i)16-s + (−0.968 − 1.67i)17-s + (1.09 + 1.88i)18-s + ⋯ |
L(s) = 1 | + (−0.300 + 0.521i)2-s + (−0.190 − 0.330i)3-s + (0.318 + 0.552i)4-s + (0.769 − 1.33i)5-s + 0.229·6-s + (−0.284 − 0.958i)7-s − 0.985·8-s + (0.427 − 0.739i)9-s + (0.463 + 0.802i)10-s + (−0.135 − 0.234i)11-s + (0.121 − 0.210i)12-s + (0.585 + 0.140i)14-s − 0.587·15-s + (−0.0222 + 0.0386i)16-s + (−0.234 − 0.406i)17-s + (0.257 + 0.445i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.168185318\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.168185318\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.751 + 2.53i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.425 - 0.737i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.330 + 0.572i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.72 + 2.98i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.448 + 0.777i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.968 + 1.67i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.519 + 0.898i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.82 - 4.89i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.83T + 29T^{2} \) |
| 31 | \( 1 + (4.56 + 7.91i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.30 - 9.17i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5.33T + 41T^{2} \) |
| 43 | \( 1 + 3.91T + 43T^{2} \) |
| 47 | \( 1 + (-3.59 + 6.22i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.69 - 8.12i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.255 + 0.442i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.718 - 1.24i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.22 + 7.31i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.44T + 71T^{2} \) |
| 73 | \( 1 + (-5.45 - 9.44i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.04 + 10.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.51T + 83T^{2} \) |
| 89 | \( 1 + (-6.80 + 11.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 0.506T + 97T^{2} \) |
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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
−9.401640595156719338884609320853, −8.757230509833018493078045922426, −7.75577638907886030690529680768, −7.15716370354588775662626678419, −6.25915381532148339571176748883, −5.55726816804658317076161255192, −4.34107595152188019954742836366, −3.39994522758848791741213359748, −1.80335420809585413847257725774, −0.52973966593910701477704331081,
1.97140952877369954648852595974, 2.38782666609851711152030244412, 3.54007927022799932610087802109, 5.11477876603124269436012616633, 5.84631269199609815280068900524, 6.52849609656611175121189507301, 7.34446488027811485440022632823, 8.666487765960607851054234557671, 9.507563159065499340622900171839, 10.15621778403875290672950647819