L(s) = 1 | + 2-s + (−0.5 − 0.866i)3-s + 4-s + (1.75 + 3.03i)5-s + (−0.5 − 0.866i)6-s + (−2.63 − 0.222i)7-s + 8-s + (−0.499 + 0.866i)9-s + (1.75 + 3.03i)10-s + (3.20 + 5.54i)11-s + (−0.5 − 0.866i)12-s + (−0.213 − 3.59i)13-s + (−2.63 − 0.222i)14-s + (1.75 − 3.03i)15-s + 16-s + 4.67·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.288 − 0.499i)3-s + 0.5·4-s + (0.784 + 1.35i)5-s + (−0.204 − 0.353i)6-s + (−0.996 − 0.0839i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.554 + 0.960i)10-s + (0.966 + 1.67i)11-s + (−0.144 − 0.249i)12-s + (−0.0590 − 0.998i)13-s + (−0.704 − 0.0593i)14-s + (0.452 − 0.784i)15-s + 0.250·16-s + 1.13·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.00881 + 0.722694i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.00881 + 0.722694i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.63 + 0.222i)T \) |
| 13 | \( 1 + (0.213 + 3.59i)T \) |
good | 5 | \( 1 + (-1.75 - 3.03i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.20 - 5.54i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 4.67T + 17T^{2} \) |
| 19 | \( 1 + (2.61 - 4.53i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.46 + 7.72i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.89T + 37T^{2} \) |
| 41 | \( 1 + (-5.09 + 8.82i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.19 + 2.06i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.44 + 4.22i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.05 - 1.82i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + (4.67 - 8.10i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.64 + 6.31i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.79 - 4.83i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.23 + 7.32i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.893 - 1.54i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.59T + 83T^{2} \) |
| 89 | \( 1 - 7.00T + 89T^{2} \) |
| 97 | \( 1 + (4.92 + 8.53i)T + (-48.5 + 84.0i)T^{2} \) |
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
−10.79827343002142325384580981354, −10.09722027179592433665635136268, −9.610745053895172106226141668199, −7.72054985109260349579645179533, −7.06077778543190808032088726083, −6.21742802771706712084153917499, −5.73642008675273007933450156477, −4.07881470209480537671837244214, −2.98498107042203867014697177399, −1.91268365663615367596141253890,
1.13218417911323628651200122569, 3.01428667893019982989718494949, 4.13780680459462548692319863473, 5.06539567021044831000545993814, 6.12335429545705566117955748740, 6.42327046264987294231018976348, 8.261557847957081455755408274663, 9.226175695029318103997723912879, 9.576474063653785975394943592798, 10.82471077461316841049547381545