L(s) = 1 | + (0.146 + 0.989i)2-s + (−0.956 + 0.290i)4-s + (0.941 − 0.336i)7-s + (−0.427 − 0.903i)8-s + (−0.671 + 0.740i)9-s + (−1.01 − 1.60i)11-s + (0.471 + 0.881i)14-s + (0.831 − 0.555i)16-s + (−0.831 − 0.555i)18-s + (1.43 − 1.24i)22-s + (0.0143 − 0.0970i)23-s + (0.970 + 0.242i)25-s + (−0.803 + 0.595i)28-s + (1.28 − 0.907i)29-s + (0.671 + 0.740i)32-s + ⋯ |
L(s) = 1 | + (0.146 + 0.989i)2-s + (−0.956 + 0.290i)4-s + (0.941 − 0.336i)7-s + (−0.427 − 0.903i)8-s + (−0.671 + 0.740i)9-s + (−1.01 − 1.60i)11-s + (0.471 + 0.881i)14-s + (0.831 − 0.555i)16-s + (−0.831 − 0.555i)18-s + (1.43 − 1.24i)22-s + (0.0143 − 0.0970i)23-s + (0.970 + 0.242i)25-s + (−0.803 + 0.595i)28-s + (1.28 − 0.907i)29-s + (0.671 + 0.740i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.184715585\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.184715585\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.146 - 0.989i)T \) |
| 7 | \( 1 + (-0.941 + 0.336i)T \) |
good | 3 | \( 1 + (0.671 - 0.740i)T^{2} \) |
| 5 | \( 1 + (-0.970 - 0.242i)T^{2} \) |
| 11 | \( 1 + (1.01 + 1.60i)T + (-0.427 + 0.903i)T^{2} \) |
| 13 | \( 1 + (0.514 - 0.857i)T^{2} \) |
| 17 | \( 1 + (-0.831 - 0.555i)T^{2} \) |
| 19 | \( 1 + (-0.803 + 0.595i)T^{2} \) |
| 23 | \( 1 + (-0.0143 + 0.0970i)T + (-0.956 - 0.290i)T^{2} \) |
| 29 | \( 1 + (-1.28 + 0.907i)T + (0.336 - 0.941i)T^{2} \) |
| 31 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 37 | \( 1 + (-1.06 + 0.0787i)T + (0.989 - 0.146i)T^{2} \) |
| 41 | \( 1 + (-0.881 + 0.471i)T^{2} \) |
| 43 | \( 1 + (-0.123 - 0.319i)T + (-0.740 + 0.671i)T^{2} \) |
| 47 | \( 1 + (0.195 + 0.980i)T^{2} \) |
| 53 | \( 1 + (0.0401 + 0.0282i)T + (0.336 + 0.941i)T^{2} \) |
| 59 | \( 1 + (-0.514 - 0.857i)T^{2} \) |
| 61 | \( 1 + (-0.0490 - 0.998i)T^{2} \) |
| 67 | \( 1 + (0.0457 + 1.86i)T + (-0.998 + 0.0490i)T^{2} \) |
| 71 | \( 1 + (-0.00961 + 0.195i)T + (-0.995 - 0.0980i)T^{2} \) |
| 73 | \( 1 + (-0.773 - 0.634i)T^{2} \) |
| 79 | \( 1 + (-1.70 - 0.168i)T + (0.980 + 0.195i)T^{2} \) |
| 83 | \( 1 + (-0.989 - 0.146i)T^{2} \) |
| 89 | \( 1 + (0.956 - 0.290i)T^{2} \) |
| 97 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.384930701542103513505333133686, −8.100388596554196922054187587715, −7.54566111820535664666228303115, −6.44355479395221887476412062307, −5.79457673116683888929519756481, −5.07527307394460430239065406828, −4.56188136455430973604507584744, −3.37563357341378819174933219127, −2.54322869438296648136467745184, −0.77897634115373715453091433857,
1.17447485124913976827740736845, 2.33381284633268643189914897923, 2.86158885991050434038168529049, 4.06951009008829820782393292988, 4.87726581450129612506823592248, 5.26628134372252742188220297766, 6.32017382938896137438884893328, 7.33989083517038951014833865360, 8.172876714663338609686119838839, 8.761811103176881726462128583030