L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.919 + 2.03i)5-s − 1.07i·7-s + 0.999i·8-s + (−0.223 − 2.22i)10-s − 0.410·11-s + (−3.30 − 1.91i)13-s + (0.537 + 0.931i)14-s + (−0.5 − 0.866i)16-s + (1.08 − 0.627i)17-s + (3.85 − 2.03i)19-s + (1.30 + 1.81i)20-s + (0.355 − 0.205i)22-s + (3.23 + 1.86i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.410 + 0.911i)5-s − 0.406i·7-s + 0.353i·8-s + (−0.0706 − 0.703i)10-s − 0.123·11-s + (−0.917 − 0.529i)13-s + (0.143 + 0.248i)14-s + (−0.125 − 0.216i)16-s + (0.263 − 0.152i)17-s + (0.884 − 0.466i)19-s + (0.291 + 0.405i)20-s + (0.0758 − 0.0437i)22-s + (0.674 + 0.389i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.052954021\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.052954021\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.919 - 2.03i)T \) |
| 19 | \( 1 + (-3.85 + 2.03i)T \) |
good | 7 | \( 1 + 1.07iT - 7T^{2} \) |
| 11 | \( 1 + 0.410T + 11T^{2} \) |
| 13 | \( 1 + (3.30 + 1.91i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.08 + 0.627i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.23 - 1.86i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.18 - 2.05i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 7.75T + 31T^{2} \) |
| 37 | \( 1 - 2.08iT - 37T^{2} \) |
| 41 | \( 1 + (-2.80 - 4.85i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.92 + 1.10i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.58 + 2.07i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.32 - 2.49i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.25 + 2.17i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.37 - 5.84i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.07 - 4.66i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.79 + 8.30i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.02 - 3.47i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.47 - 7.74i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.12iT - 83T^{2} \) |
| 89 | \( 1 + (-2.72 + 4.72i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.5 + 6.08i)T + (48.5 - 84.0i)T^{2} \) |
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
−9.588405667335765060994937129175, −8.547697962996287313862229719000, −7.61502481572123268816574012484, −7.30838082568864974511676622476, −6.50074113604867797119192092951, −5.48583111977840603322671561978, −4.58044580630981911594557921571, −3.29252351402431800654679783911, −2.53729076708864159163825586652, −0.860076557159579192959468012616,
0.70242771721269930010662762432, 1.96775897276937403729343375485, 3.07168530394494723123333613454, 4.21605897117625013680910342432, 5.02216373462855210425972537054, 5.95172648759059984177222952677, 7.12902767957916820515620043076, 7.79006270359780795190070248521, 8.526303206399660240886351636795, 9.255203294270053757488771277328