| L(s) = 1 | + (0.626 − 1.92i)2-s + (3.14 + 2.28i)4-s + (3.12 − 10.7i)5-s − 16.4·7-s + (19.4 − 14.1i)8-s + (−18.7 − 12.7i)10-s + (1.41 − 4.34i)11-s + (−17.0 − 52.4i)13-s + (−10.2 + 31.6i)14-s + (−5.47 − 16.8i)16-s + (1.72 − 1.25i)17-s + (22.2 − 16.1i)19-s + (34.3 − 26.6i)20-s + (−7.48 − 5.43i)22-s + (30.4 − 93.5i)23-s + ⋯ |
| L(s) = 1 | + (0.221 − 0.681i)2-s + (0.393 + 0.285i)4-s + (0.279 − 0.960i)5-s − 0.887·7-s + (0.861 − 0.626i)8-s + (−0.592 − 0.402i)10-s + (0.0386 − 0.119i)11-s + (−0.363 − 1.11i)13-s + (−0.196 + 0.605i)14-s + (−0.0855 − 0.263i)16-s + (0.0246 − 0.0179i)17-s + (0.268 − 0.194i)19-s + (0.384 − 0.298i)20-s + (−0.0725 − 0.0527i)22-s + (0.275 − 0.848i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.905031 - 1.77786i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.905031 - 1.77786i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-3.12 + 10.7i)T \) |
| good | 2 | \( 1 + (-0.626 + 1.92i)T + (-6.47 - 4.70i)T^{2} \) |
| 7 | \( 1 + 16.4T + 343T^{2} \) |
| 11 | \( 1 + (-1.41 + 4.34i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (17.0 + 52.4i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-1.72 + 1.25i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-22.2 + 16.1i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-30.4 + 93.5i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (2.70 + 1.96i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-54.6 + 39.6i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (56.4 + 173. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (11.7 + 36.2i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 252.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-35.6 - 25.9i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-163. - 119. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (36.4 + 112. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (248. - 763. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-523. + 380. i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-798. - 580. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (256. - 790. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-822. - 597. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-926. + 673. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-349. + 1.07e3i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-457. - 332. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−11.64220004252767013290836072631, −10.46795201742487937778743583534, −9.751394521666883618516241595954, −8.583498751636765359633232426955, −7.46712044947569818626795389558, −6.24223384352248283250893711457, −4.96855282456264237702439546587, −3.61421469179315300196325582066, −2.44248455446378461882824082611, −0.74906893116458237828877667408,
1.96158774434518291132192230721, 3.38119536389337596486102898373, 5.05452183149533987545343734346, 6.34453778286704020585413687180, 6.77327726716246141470106858792, 7.76726008287614647068677968586, 9.397929500577443838798197583260, 10.14216922154138931868068274434, 11.15151045971947176937732895634, 12.01321379618311926796574435839
