L(s) = 1 | + (1.05 + 0.769i)3-s + (−0.809 + 0.587i)4-s + (−0.592 + 1.82i)5-s + (0.221 + 0.680i)9-s − 1.30·12-s + (−2.03 + 1.47i)15-s + (0.309 − 0.951i)16-s + (−0.592 − 1.82i)20-s + 0.830·23-s + (−2.17 − 1.57i)25-s + (0.115 − 0.354i)27-s + (0.256 + 0.790i)31-s + (−0.578 − 0.420i)36-s + (0.230 − 0.167i)37-s − 1.37·45-s + ⋯ |
L(s) = 1 | + (1.05 + 0.769i)3-s + (−0.809 + 0.587i)4-s + (−0.592 + 1.82i)5-s + (0.221 + 0.680i)9-s − 1.30·12-s + (−2.03 + 1.47i)15-s + (0.309 − 0.951i)16-s + (−0.592 − 1.82i)20-s + 0.830·23-s + (−2.17 − 1.57i)25-s + (0.115 − 0.354i)27-s + (0.256 + 0.790i)31-s + (−0.578 − 0.420i)36-s + (0.230 − 0.167i)37-s − 1.37·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.079491919\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.079491919\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (-1.05 - 0.769i)T + (0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (0.592 - 1.82i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 - 0.830T + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.256 - 0.790i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.230 + 0.167i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.230 - 0.167i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.519 - 1.60i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (1.36 - 0.988i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + 1.91T + T^{2} \) |
| 71 | \( 1 + (0.0879 - 0.270i)T + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 - 1.68T + T^{2} \) |
| 97 | \( 1 + (-0.256 - 0.790i)T + (-0.809 + 0.587i)T^{2} \) |
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
−10.18575820798024583986580857047, −9.207421727153266542853215314726, −8.653844461737907234749489494118, −7.69774549632014540857919070169, −7.24165769224363683609977769040, −6.12661232572207321245769967708, −4.66635500630193628610848397838, −3.87839042010484509488721089841, −3.18400345339362882915549145133, −2.65620714286417271687685477690,
0.872889831212535254413303641556, 1.86561170943081062170291618874, 3.44118575946111986124408241879, 4.47338998128706310519768833091, 5.06498667333508527954266053799, 6.07905194623091371201020407181, 7.46677221903042754653087295699, 8.049344426818605420536100900692, 8.803131315784967704744015848869, 9.076589757897628794789549067916