L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + 0.999·8-s − 0.999·10-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)20-s + (0.499 − 0.866i)22-s − 0.999·28-s + (−1 − 1.73i)29-s + (0.5 − 0.866i)31-s + (−0.499 + 0.866i)32-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + 0.999·8-s − 0.999·10-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)20-s + (0.499 − 0.866i)22-s − 0.999·28-s + (−1 − 1.73i)29-s + (0.5 − 0.866i)31-s + (−0.499 + 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8128726147\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8128726147\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−10.58483281239145764204677943832, −9.518896883150050417040229457045, −9.239818213571234348258372061148, −8.293055369902239843732043944791, −7.47139026214265691637763320037, −5.99832581164244129955736581992, −4.93683633261307930000651656722, −4.09646608430318677494686539128, −2.47453976508911770245719311557, −1.54583924612058021690674937418,
1.48015200185724014076485780921, 3.32227466240535276856593802975, 4.62121139574856283573157873576, 5.72291631023388105054353656416, 6.62538321330765683539445769495, 7.22940051577507861735278782284, 8.213548190356932383142243508134, 9.060755381578823709194743200525, 10.00822578464756570402032991624, 10.79448581127640307967298437489