L(s) = 1 | + (1.26 − 0.636i)2-s + (0.577 − 1.63i)3-s + (1.18 − 1.60i)4-s + (−0.161 + 0.442i)5-s + (−0.311 − 2.42i)6-s + (0.984 + 0.173i)7-s + (0.478 − 2.78i)8-s + (−2.33 − 1.88i)9-s + (0.0783 + 0.661i)10-s + (−2.51 + 0.914i)11-s + (−1.93 − 2.87i)12-s + (0.804 − 0.675i)13-s + (1.35 − 0.407i)14-s + (0.629 + 0.518i)15-s + (−1.17 − 3.82i)16-s + (5.28 − 3.05i)17-s + ⋯ |
L(s) = 1 | + (0.892 − 0.450i)2-s + (0.333 − 0.942i)3-s + (0.594 − 0.803i)4-s + (−0.0720 + 0.197i)5-s + (−0.126 − 0.991i)6-s + (0.372 + 0.0656i)7-s + (0.169 − 0.985i)8-s + (−0.778 − 0.628i)9-s + (0.0247 + 0.209i)10-s + (−0.757 + 0.275i)11-s + (−0.559 − 0.828i)12-s + (0.223 − 0.187i)13-s + (0.361 − 0.108i)14-s + (0.162 + 0.133i)15-s + (−0.292 − 0.956i)16-s + (1.28 − 0.739i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43396 - 2.39603i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43396 - 2.39603i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.26 + 0.636i)T \) |
| 3 | \( 1 + (-0.577 + 1.63i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
good | 5 | \( 1 + (0.161 - 0.442i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (2.51 - 0.914i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.804 + 0.675i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.28 + 3.05i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.94 + 1.12i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.02 + 5.81i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.485 + 0.578i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (5.63 - 0.992i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-5.10 - 8.84i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.917 - 1.09i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-4.06 - 11.1i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.0862 + 0.489i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 9.90iT - 53T^{2} \) |
| 59 | \( 1 + (-9.24 - 3.36i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.06 + 6.05i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.34 - 6.37i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.25 + 3.91i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.88 + 13.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.32 - 3.96i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (10.1 + 8.49i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.16 + 1.25i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.27 - 1.19i)T + (74.3 - 62.3i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.29953930473600036976828116535, −9.286347185761894673380102438548, −8.099694680960466404674327970630, −7.38341825772024253432131352321, −6.47753912029319828593746470390, −5.56515898916718409231836513197, −4.61080544414862217640337908313, −3.18509411878324761249375713594, −2.47906306801800154417359637048, −1.09997491454847187409806004979,
2.23772525848881932471282230374, 3.53233108755951260512584738406, 4.13487169686813095756813579434, 5.40790745483569205322456780306, 5.67408623659453039506436907083, 7.22144383975311511557329016115, 8.093210339278339753461369466352, 8.647077971007374702169855254882, 9.854178849813803806320311557391, 10.76537742955053628201091216635