L(s) = 1 | + (−0.909 + 0.909i)2-s + 2.34i·4-s + 4.16i·5-s − 9.68·7-s + (−5.77 − 5.77i)8-s + (−3.78 − 3.78i)10-s + (0.334 − 0.334i)11-s − 12.2i·13-s + (8.81 − 8.81i)14-s + 1.11·16-s + (−6.80 + 6.80i)17-s + (14.6 − 14.6i)19-s − 9.76·20-s + 0.609i·22-s + 10.0·23-s + ⋯ |
L(s) = 1 | + (−0.454 + 0.454i)2-s + 0.586i·4-s + 0.832i·5-s − 1.38·7-s + (−0.721 − 0.721i)8-s + (−0.378 − 0.378i)10-s + (0.0304 − 0.0304i)11-s − 0.940i·13-s + (0.629 − 0.629i)14-s + 0.0698·16-s + (−0.400 + 0.400i)17-s + (0.771 − 0.771i)19-s − 0.488·20-s + 0.0276i·22-s + 0.438·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0435979 - 0.0686723i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0435979 - 0.0686723i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 29 | \( 1 + (28.1 + 7.15i)T \) |
good | 2 | \( 1 + (0.909 - 0.909i)T - 4iT^{2} \) |
| 5 | \( 1 - 4.16iT - 25T^{2} \) |
| 7 | \( 1 + 9.68T + 49T^{2} \) |
| 11 | \( 1 + (-0.334 + 0.334i)T - 121iT^{2} \) |
| 13 | \( 1 + 12.2iT - 169T^{2} \) |
| 17 | \( 1 + (6.80 - 6.80i)T - 289iT^{2} \) |
| 19 | \( 1 + (-14.6 + 14.6i)T - 361iT^{2} \) |
| 23 | \( 1 - 10.0T + 529T^{2} \) |
| 31 | \( 1 + (37.3 - 37.3i)T - 961iT^{2} \) |
| 37 | \( 1 + (45.0 + 45.0i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (22.8 + 22.8i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (17.5 - 17.5i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (2.20 + 2.20i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + 90.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 90.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + (29.4 - 29.4i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 - 31.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 99.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (3.96 + 3.96i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + (40.3 - 40.3i)T - 6.24e3iT^{2} \) |
| 83 | \( 1 + 137.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (83.1 - 83.1i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (79.9 + 79.9i)T + 9.40e3iT^{2} \) |
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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
−12.72933099684857815743132968999, −11.34394819240800989857100417053, −10.36897818920995769141282041686, −9.418749642238232688572028844251, −8.584365998975770390441674698300, −7.18824332127601319709501660466, −6.87874395260221176241824337663, −5.61982907138171082196535467777, −3.59267828356608614895019747818, −2.92557861774182032329452598611,
0.04685008673065721138079459684, 1.67011014177566652360565993152, 3.34066383667063266520288395341, 4.91711141613747991923835909870, 6.00697274747154017506562947011, 7.04163460003060300376984296267, 8.625532280329793384822439697273, 9.426054619247798178789213277889, 9.846562679565378424984492349740, 11.09191756200755102613624881142