L(s) = 1 | + (1.35 − 0.760i)2-s + (−0.953 + 0.302i)3-s + (0.731 − 1.20i)4-s + (−1.05 + 1.13i)6-s + (0.0215 − 0.630i)8-s + (0.816 − 0.576i)9-s + (−0.333 + 1.36i)12-s + (0.196 + 0.379i)16-s + (1.30 + 1.49i)17-s + (0.666 − 1.40i)18-s + (−0.806 − 0.655i)19-s + (0.264 − 0.470i)23-s + (0.170 + 0.607i)24-s + (−0.604 + 0.796i)27-s + (1.77 − 0.917i)31-s + (1.08 + 0.709i)32-s + ⋯ |
L(s) = 1 | + (1.35 − 0.760i)2-s + (−0.953 + 0.302i)3-s + (0.731 − 1.20i)4-s + (−1.05 + 1.13i)6-s + (0.0215 − 0.630i)8-s + (0.816 − 0.576i)9-s + (−0.333 + 1.36i)12-s + (0.196 + 0.379i)16-s + (1.30 + 1.49i)17-s + (0.666 − 1.40i)18-s + (−0.806 − 0.655i)19-s + (0.264 − 0.470i)23-s + (0.170 + 0.607i)24-s + (−0.604 + 0.796i)27-s + (1.77 − 0.917i)31-s + (1.08 + 0.709i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.078139361\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.078139361\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.953 - 0.302i)T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + (0.930 - 0.366i)T \) |
good | 2 | \( 1 + (-1.35 + 0.760i)T + (0.519 - 0.854i)T^{2} \) |
| 7 | \( 1 + (-0.269 + 0.962i)T^{2} \) |
| 11 | \( 1 + (-0.990 + 0.136i)T^{2} \) |
| 13 | \( 1 + (0.631 - 0.775i)T^{2} \) |
| 17 | \( 1 + (-1.30 - 1.49i)T + (-0.136 + 0.990i)T^{2} \) |
| 19 | \( 1 + (0.806 + 0.655i)T + (0.203 + 0.979i)T^{2} \) |
| 23 | \( 1 + (-0.264 + 0.470i)T + (-0.519 - 0.854i)T^{2} \) |
| 29 | \( 1 + (0.775 - 0.631i)T^{2} \) |
| 31 | \( 1 + (-1.77 + 0.917i)T + (0.576 - 0.816i)T^{2} \) |
| 37 | \( 1 + (0.730 + 0.682i)T^{2} \) |
| 41 | \( 1 + (-0.0682 - 0.997i)T^{2} \) |
| 43 | \( 1 + (-0.887 + 0.460i)T^{2} \) |
| 53 | \( 1 + (-1.15 + 0.0393i)T + (0.997 - 0.0682i)T^{2} \) |
| 59 | \( 1 + (0.460 - 0.887i)T^{2} \) |
| 61 | \( 1 + (0.614 + 0.266i)T + (0.682 + 0.730i)T^{2} \) |
| 67 | \( 1 + (-0.269 - 0.962i)T^{2} \) |
| 71 | \( 1 + (-0.854 + 0.519i)T^{2} \) |
| 73 | \( 1 + (0.942 - 0.334i)T^{2} \) |
| 79 | \( 1 + (-0.133 + 0.0277i)T + (0.917 - 0.398i)T^{2} \) |
| 83 | \( 1 + (0.897 - 1.02i)T + (-0.136 - 0.990i)T^{2} \) |
| 89 | \( 1 + (0.203 - 0.979i)T^{2} \) |
| 97 | \( 1 + (0.816 - 0.576i)T^{2} \) |
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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
−8.676824064393406921749790237334, −7.902491364529819697578370272272, −6.68814256170241223845671763499, −6.14653109425276467423940574915, −5.51579516808546436563816981877, −4.72020616379033763866283047841, −4.13526781420250541123315705954, −3.40562284707297623377374337249, −2.33993961242287828684944976471, −1.19138686118125022824304989020,
1.14198404773896088731368285856, 2.69645090102621259422296264784, 3.61561418400725893898742428291, 4.63348318173944061653861482018, 5.04380126279816932833266300782, 5.84906329114394329646499828728, 6.33785436664193061329008181976, 7.19279250228381730447878103343, 7.57120894336298850569297580235, 8.525191062923515711847082544612