L(s) = 1 | + (−0.516 − 1.31i)2-s + 1.28·3-s + (−1.46 + 1.36i)4-s + (2.07 − 0.841i)5-s + (−0.662 − 1.68i)6-s + (−1.13 − 1.13i)7-s + (2.54 + 1.22i)8-s − 1.35·9-s + (−2.17 − 2.29i)10-s + (−2.32 + 2.32i)11-s + (−1.87 + 1.74i)12-s − 1.36i·13-s + (−0.911 + 2.08i)14-s + (2.65 − 1.07i)15-s + (0.297 − 3.98i)16-s + (5.25 + 5.25i)17-s + ⋯ |
L(s) = 1 | + (−0.365 − 0.930i)2-s + 0.739·3-s + (−0.732 + 0.680i)4-s + (0.926 − 0.376i)5-s + (−0.270 − 0.688i)6-s + (−0.430 − 0.430i)7-s + (0.901 + 0.433i)8-s − 0.452·9-s + (−0.688 − 0.724i)10-s + (−0.700 + 0.700i)11-s + (−0.542 + 0.503i)12-s − 0.378i·13-s + (−0.243 + 0.558i)14-s + (0.685 − 0.278i)15-s + (0.0744 − 0.997i)16-s + (1.27 + 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.820985 - 0.502484i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.820985 - 0.502484i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.516 + 1.31i)T \) |
| 5 | \( 1 + (-2.07 + 0.841i)T \) |
good | 3 | \( 1 - 1.28T + 3T^{2} \) |
| 7 | \( 1 + (1.13 + 1.13i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.32 - 2.32i)T - 11iT^{2} \) |
| 13 | \( 1 + 1.36iT - 13T^{2} \) |
| 17 | \( 1 + (-5.25 - 5.25i)T + 17iT^{2} \) |
| 19 | \( 1 + (3.69 - 3.69i)T - 19iT^{2} \) |
| 23 | \( 1 + (0.911 - 0.911i)T - 23iT^{2} \) |
| 29 | \( 1 + (2.37 + 2.37i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.242iT - 31T^{2} \) |
| 37 | \( 1 + 3.34iT - 37T^{2} \) |
| 41 | \( 1 - 2.66iT - 41T^{2} \) |
| 43 | \( 1 + 9.04iT - 43T^{2} \) |
| 47 | \( 1 + (-7.87 + 7.87i)T - 47iT^{2} \) |
| 53 | \( 1 + 5.80T + 53T^{2} \) |
| 59 | \( 1 + (-5.91 - 5.91i)T + 59iT^{2} \) |
| 61 | \( 1 + (6.67 - 6.67i)T - 61iT^{2} \) |
| 67 | \( 1 + 4.54iT - 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + (1.49 + 1.49i)T + 73iT^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 3.26T + 83T^{2} \) |
| 89 | \( 1 - 9.77T + 89T^{2} \) |
| 97 | \( 1 + (1.63 + 1.63i)T + 97iT^{2} \) |
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
−13.92296499365289449809594290116, −13.03696507859897265674346255587, −12.29389435586936793981524525252, −10.46946280628922074534058066252, −9.955346989638666686820910924920, −8.734604720205434827835677184253, −7.77544458698373091190194800614, −5.62371777204037166138815658596, −3.71502546718775458421505513085, −2.12465614339622644752936543920,
2.83849966474416559970015879626, 5.30725364707375854718797864753, 6.37219222552418291322396045844, 7.77983732201985659571703268865, 8.987738202986269881935135407286, 9.662962407232584177181099776183, 10.99350542815591883765181350326, 12.94128656867917604875058299039, 13.94591109732664185281184535238, 14.43007696586270425684229907165