L(s) = 1 | + (−1.22 − 2.12i)2-s + (−1.71 + 0.228i)3-s + (−1.99 + 3.45i)4-s + (0.5 − 0.866i)5-s + (2.58 + 3.36i)6-s + (1.96 + 3.40i)7-s + 4.88·8-s + (2.89 − 0.784i)9-s − 2.44·10-s + (−2.33 − 4.04i)11-s + (2.63 − 6.39i)12-s + (0.5 − 0.866i)13-s + (4.81 − 8.34i)14-s + (−0.660 + 1.60i)15-s + (−1.98 − 3.43i)16-s − 7.96·17-s + ⋯ |
L(s) = 1 | + (−0.865 − 1.49i)2-s + (−0.991 + 0.131i)3-s + (−0.998 + 1.72i)4-s + (0.223 − 0.387i)5-s + (1.05 + 1.37i)6-s + (0.743 + 1.28i)7-s + 1.72·8-s + (0.965 − 0.261i)9-s − 0.774·10-s + (−0.704 − 1.22i)11-s + (0.761 − 1.84i)12-s + (0.138 − 0.240i)13-s + (1.28 − 2.22i)14-s + (−0.170 + 0.413i)15-s + (−0.495 − 0.858i)16-s − 1.93·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0695993 + 0.109573i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0695993 + 0.109573i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.71 - 0.228i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (1.22 + 2.12i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-1.96 - 3.40i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.33 + 4.04i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 7.96T + 17T^{2} \) |
| 19 | \( 1 + 0.120T + 19T^{2} \) |
| 23 | \( 1 + (0.164 - 0.284i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.136 - 0.236i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.28 + 5.69i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.43T + 37T^{2} \) |
| 41 | \( 1 + (0.0455 - 0.0788i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.89 + 6.75i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.78 - 10.0i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + (2.77 - 4.81i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.63 + 11.4i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.41 - 5.90i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.32T + 71T^{2} \) |
| 73 | \( 1 + 7.37T + 73T^{2} \) |
| 79 | \( 1 + (-5.95 - 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.774 + 1.34i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3.59T + 89T^{2} \) |
| 97 | \( 1 + (2.63 + 4.57i)T + (-48.5 + 84.0i)T^{2} \) |
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
−10.42408203605097420891711877039, −9.262128413283976289918944186797, −8.739990914526727155855018996626, −7.948512514550541430771809685528, −6.24354695114939431446216780113, −5.33269963575003673882393912929, −4.31332992313106294758834331250, −2.77391201707971928547022100710, −1.69305604000094333125267356719, −0.11133264065886383651619888157,
1.64884253743066931481331033992, 4.49689393838767806751401622992, 4.94919985313441949329677127178, 6.27478195199420452738566390663, 6.99741103920847296049086853832, 7.36914608131409111749147972065, 8.386486086512732800867318986734, 9.528559935535686918978650739963, 10.53351157521946458468975790416, 10.68161576479550778844807752016