L(s) = 1 | + (−0.841 − 1.45i)2-s + (0.327 − 0.566i)3-s + (−0.915 + 1.58i)4-s − 1.10·6-s + (0.235 + 0.971i)7-s + 1.39·8-s + (0.286 + 0.495i)9-s + (−0.580 + 1.00i)11-s + (0.598 + 1.03i)12-s + (1.21 − 1.16i)14-s + (−0.260 − 0.451i)16-s + (0.481 − 0.833i)18-s + (0.142 + 0.246i)19-s + (0.627 + 0.184i)21-s + 1.95·22-s + ⋯ |
L(s) = 1 | + (−0.841 − 1.45i)2-s + (0.327 − 0.566i)3-s + (−0.915 + 1.58i)4-s − 1.10·6-s + (0.235 + 0.971i)7-s + 1.39·8-s + (0.286 + 0.495i)9-s + (−0.580 + 1.00i)11-s + (0.598 + 1.03i)12-s + (1.21 − 1.16i)14-s + (−0.260 − 0.451i)16-s + (0.481 − 0.833i)18-s + (0.142 + 0.246i)19-s + (0.627 + 0.184i)21-s + 1.95·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7060091857\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7060091857\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.235 - 0.971i)T \) |
| 167 | \( 1 - T \) |
good | 2 | \( 1 + (0.841 + 1.45i)T + (-0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.142 - 0.246i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - 1.44T + T^{2} \) |
| 31 | \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.928 + 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.654 - 1.13i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.981 + 1.70i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - 0.830T + T^{2} \) |
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
−10.03181997616453620620039406967, −9.148181748973727396363194994342, −8.377110261816994443409547544534, −7.82561351502865848925015000883, −6.87521129584792833035567313012, −5.44590393304645211919282225380, −4.45811693492971627351358881632, −3.09259517337383944805934376050, −2.25861590778850089877773312935, −1.58083986465468025211141020362,
0.868664887390317409391949395980, 3.09145810508611978222108611565, 4.26441878822654045465014193459, 5.11888559738992185465042127336, 6.24863929625052066737878913444, 6.78671700034018753203998635453, 7.85564045373644864767153114335, 8.256993618121394650619081418590, 9.121269964901129002753871314348, 9.900585356420569185836902161496