L(s) = 1 | + (−0.994 + 0.104i)2-s + (0.258 − 0.965i)3-s + (0.978 − 0.207i)4-s + (0.743 − 0.669i)5-s + (−0.156 + 0.987i)6-s + (−0.951 + 0.309i)8-s + (−0.866 − 0.5i)9-s + (−0.669 + 0.743i)10-s + (0.998 − 0.0523i)11-s + (0.0523 − 0.998i)12-s + (−0.987 − 0.156i)13-s + (−0.453 − 0.891i)15-s + (0.913 − 0.406i)16-s + (−0.0523 − 0.998i)17-s + (0.913 + 0.406i)18-s + (0.358 − 0.933i)19-s + ⋯ |
L(s) = 1 | + (−0.994 + 0.104i)2-s + (0.258 − 0.965i)3-s + (0.978 − 0.207i)4-s + (0.743 − 0.669i)5-s + (−0.156 + 0.987i)6-s + (−0.951 + 0.309i)8-s + (−0.866 − 0.5i)9-s + (−0.669 + 0.743i)10-s + (0.998 − 0.0523i)11-s + (0.0523 − 0.998i)12-s + (−0.987 − 0.156i)13-s + (−0.453 − 0.891i)15-s + (0.913 − 0.406i)16-s + (−0.0523 − 0.998i)17-s + (0.913 + 0.406i)18-s + (0.358 − 0.933i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5159731902 - 0.7648490409i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5159731902 - 0.7648490409i\) |
\(L(1)\) |
\(\approx\) |
\(0.7295579147 - 0.4114944288i\) |
\(L(1)\) |
\(\approx\) |
\(0.7295579147 - 0.4114944288i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (-0.994 + 0.104i)T \) |
| 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 + (0.743 - 0.669i)T \) |
| 11 | \( 1 + (0.998 - 0.0523i)T \) |
| 13 | \( 1 + (-0.987 - 0.156i)T \) |
| 17 | \( 1 + (-0.0523 - 0.998i)T \) |
| 19 | \( 1 + (0.358 - 0.933i)T \) |
| 23 | \( 1 + (0.104 + 0.994i)T \) |
| 29 | \( 1 + (-0.891 + 0.453i)T \) |
| 31 | \( 1 + (0.669 - 0.743i)T \) |
| 37 | \( 1 + (0.669 + 0.743i)T \) |
| 43 | \( 1 + (0.587 + 0.809i)T \) |
| 47 | \( 1 + (-0.777 - 0.629i)T \) |
| 53 | \( 1 + (-0.838 - 0.544i)T \) |
| 59 | \( 1 + (-0.913 - 0.406i)T \) |
| 61 | \( 1 + (0.406 + 0.913i)T \) |
| 67 | \( 1 + (0.544 - 0.838i)T \) |
| 71 | \( 1 + (0.453 - 0.891i)T \) |
| 73 | \( 1 + (0.866 - 0.5i)T \) |
| 79 | \( 1 + (-0.965 + 0.258i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.933 + 0.358i)T \) |
| 97 | \( 1 + (0.453 + 0.891i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−26.09657017300693702887246380505, −25.12930123694810455148602257574, −24.54096032517302279893645721226, −22.76286134469124068605590287504, −21.93904798351408300714666263326, −21.269417584949524386876532728772, −20.30535906878261021291578955315, −19.43667304749731995577830889848, −18.613859545938272730554127322138, −17.19634395878929647048942517771, −17.07148546375330555483069114938, −15.828737961314781462346266187781, −14.66386686037241150226396157722, −14.34920108248458902132751993354, −12.53815923462042011926702576788, −11.37094073343589550298064850114, −10.46160301822694353844583293383, −9.77476231337997568114015302015, −9.05381836053039758046163232198, −7.903184056358832768903414939428, −6.64959385028109075694203035091, −5.70349318546770508186101577509, −4.04101423470052994136066397704, −2.86229853198944251811400031696, −1.80097367571633047318303847251,
0.83017783914537736581185267282, 1.889716450155919487880707675975, 2.95164204794783810805477742244, 5.11224755437023352220142781358, 6.264248679559538467584487306460, 7.14344828305915567026214910769, 8.07428391703628458199447399666, 9.382604089287185073474093926583, 9.47567584683321634490995250084, 11.32539484029113064410112353554, 12.02040361954293455621113177197, 13.122483558373254329180685422889, 14.11026591694769857350637490166, 15.11105995596073343724927914311, 16.48307940862470460821167632800, 17.28482731736893749702371775771, 17.80326594313492484314375125708, 18.81958263098839362473030867334, 19.84515588813107092701003614117, 20.20166775801792919664074832084, 21.40931930578861825957103587011, 22.58222220894534144926144869774, 24.13495908454283797520886293381, 24.43849435547400546006010887689, 25.255197759473152124415930614622