L(s) = 1 | + 3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)7-s + 9-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s − 19-s + (−0.5 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 27-s + 0.999·28-s + (−1 + 1.73i)31-s + (−0.5 − 0.866i)36-s + (0.5 − 0.866i)37-s + ⋯ |
L(s) = 1 | + 3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)7-s + 9-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s − 19-s + (−0.5 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 27-s + 0.999·28-s + (−1 + 1.73i)31-s + (−0.5 − 0.866i)36-s + (0.5 − 0.866i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8716818788\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8716818788\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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
−12.47280104986919417425236328904, −10.85357918957215016724031437838, −10.00632972835089425287224747310, −9.167265071928771324305868643822, −8.554337223331167530883043376707, −7.27257749214163409643245943034, −5.98868748900753155231223910514, −4.94176371987835500257443711914, −3.49716115974458044202166729864, −2.08137434854167615783808386896,
2.46618336311894048396802786649, 3.85365541895090556900335790972, 4.43874307070481477132483183021, 6.56356817623343802398847747667, 7.51989052799530073639003036007, 8.269488523470644085255250662983, 9.345112294070437519236864962480, 9.927840296809085645880369734637, 11.28312144833772637619063067850, 12.53134817347494461545917938812