L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.00696 − 2.23i)5-s − 0.999i·6-s + (3.60 − 2.08i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (1.93 + 1.11i)10-s + 5.62·11-s + (0.866 + 0.499i)12-s + (−2.80 − 4.85i)13-s + 4.16i·14-s + (1.12 + 1.93i)15-s + (−0.5 + 0.866i)16-s + (1.43 − 2.48i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.00311 − 0.999i)5-s − 0.408i·6-s + (1.36 − 0.786i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.613 + 0.351i)10-s + 1.69·11-s + (0.249 + 0.144i)12-s + (−0.777 − 1.34i)13-s + 1.11i·14-s + (0.290 + 0.499i)15-s + (−0.125 + 0.216i)16-s + (0.347 − 0.601i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.147869173\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.147869173\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.00696 + 2.23i)T \) |
| 37 | \( 1 + (2.01 - 5.73i)T \) |
good | 7 | \( 1 + (-3.60 + 2.08i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 + (2.80 + 4.85i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.43 + 2.48i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.16 - 2.98i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 + 9.90iT - 29T^{2} \) |
| 31 | \( 1 - 4.33iT - 31T^{2} \) |
| 41 | \( 1 + (0.470 + 0.815i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 4.29T + 43T^{2} \) |
| 47 | \( 1 - 0.365iT - 47T^{2} \) |
| 53 | \( 1 + (3.87 + 2.23i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-9.52 - 5.49i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.20 + 1.27i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.92 + 1.10i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.08 + 10.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 11.5iT - 73T^{2} \) |
| 79 | \( 1 + (8.57 - 4.94i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.157 - 0.0907i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.38 + 4.26i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8.96T + 97T^{2} \) |
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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
−9.851277585336465056967528360913, −8.646395336098065900847723157151, −8.108367057471691246820488727876, −7.36692417825053833973973017897, −6.21924930384098846036022924086, −5.43447106783200435078701799538, −4.51348885044129364766473399357, −4.02895951426498882965503638098, −1.69926449144984937485077906945, −0.63857483576614362832513263253,
1.70508828136092129350812405116, 2.20592195358077259406201889731, 3.86109148943630210045453327645, 4.60130739777583459947981327757, 5.89431475529477387019386931698, 6.72867105764554285512272755026, 7.45383196010878141055787366962, 8.548749023388937385411486319448, 9.142413576135216297569865508781, 10.12083846926877711238840142059