L(s) = 1 | + (0.161 − 0.0930i)2-s + (−0.5 + 0.866i)3-s + (−0.982 + 1.70i)4-s + (−2.28 + 1.31i)5-s + 0.186i·6-s + (0.161 − 2.64i)7-s + 0.738i·8-s + (−0.499 − 0.866i)9-s + (−0.245 + 0.425i)10-s + (−1.56 − 0.904i)11-s + (−0.982 − 1.70i)12-s + (−2.45 − 2.63i)13-s + (−0.219 − 0.440i)14-s − 2.63i·15-s + (−1.89 − 3.28i)16-s + (−3.13 + 5.42i)17-s + ⋯ |
L(s) = 1 | + (0.113 − 0.0658i)2-s + (−0.288 + 0.499i)3-s + (−0.491 + 0.851i)4-s + (−1.02 + 0.590i)5-s + 0.0759i·6-s + (0.0609 − 0.998i)7-s + 0.260i·8-s + (−0.166 − 0.288i)9-s + (−0.0776 + 0.134i)10-s + (−0.472 − 0.272i)11-s + (−0.283 − 0.491i)12-s + (−0.681 − 0.731i)13-s + (−0.0587 − 0.117i)14-s − 0.681i·15-s + (−0.474 − 0.821i)16-s + (−0.760 + 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0151703 - 0.298256i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0151703 - 0.298256i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.161 + 2.64i)T \) |
| 13 | \( 1 + (2.45 + 2.63i)T \) |
good | 2 | \( 1 + (-0.161 + 0.0930i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (2.28 - 1.31i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.56 + 0.904i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (3.13 - 5.42i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.44 - 1.41i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.12 - 3.68i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 0.456T + 29T^{2} \) |
| 31 | \( 1 + (-2.76 - 1.59i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.00 - 1.73i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 7.18iT - 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 + (10.3 - 5.95i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.23 + 3.87i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (8.83 + 5.09i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.59 + 7.95i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.40 + 0.810i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.85iT - 71T^{2} \) |
| 73 | \( 1 + (-12.3 - 7.15i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.86 - 10.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 11.7iT - 83T^{2} \) |
| 89 | \( 1 + (0.0791 - 0.0457i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 1.62iT - 97T^{2} \) |
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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
−12.43705358685005767849642377094, −11.23736484546747680214167426007, −10.77403636021559299893408787719, −9.688372205731021124120054272233, −8.222453744395989572047177681238, −7.77271458959824161315704207057, −6.60820397104374629022543287485, −4.90956532580970360417078319182, −3.96868132989826232850218258348, −3.16392909525715900287270507496,
0.22163788301526042071666887973, 2.31054175877019514651991694868, 4.52027136567430374991273778869, 5.05982502072571631395988108568, 6.36761818870154946593675498890, 7.43517409898615306884637938185, 8.695876703629953123179387993140, 9.248754321659854059421290380425, 10.59937563150778394200965801332, 11.65725742141290686420162601519