L(s) = 1 | + (0.559 + 0.829i)2-s + (0.0348 + 0.999i)3-s + (−0.374 + 0.927i)4-s + (0.438 + 0.898i)5-s + (−0.809 + 0.587i)6-s + (0.990 − 0.139i)7-s + (−0.978 + 0.207i)8-s + (−0.997 + 0.0697i)9-s + (−0.5 + 0.866i)10-s + (−0.939 − 0.342i)12-s + (−0.241 − 0.970i)13-s + (0.669 + 0.743i)14-s + (−0.882 + 0.469i)15-s + (−0.719 − 0.694i)16-s + (−0.241 + 0.970i)17-s + (−0.615 − 0.788i)18-s + ⋯ |
L(s) = 1 | + (0.559 + 0.829i)2-s + (0.0348 + 0.999i)3-s + (−0.374 + 0.927i)4-s + (0.438 + 0.898i)5-s + (−0.809 + 0.587i)6-s + (0.990 − 0.139i)7-s + (−0.978 + 0.207i)8-s + (−0.997 + 0.0697i)9-s + (−0.5 + 0.866i)10-s + (−0.939 − 0.342i)12-s + (−0.241 − 0.970i)13-s + (0.669 + 0.743i)14-s + (−0.882 + 0.469i)15-s + (−0.719 − 0.694i)16-s + (−0.241 + 0.970i)17-s + (−0.615 − 0.788i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1316218208 + 1.755486184i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1316218208 + 1.755486184i\) |
\(L(1)\) |
\(\approx\) |
\(0.7414190668 + 1.230823450i\) |
\(L(1)\) |
\(\approx\) |
\(0.7414190668 + 1.230823450i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (0.559 + 0.829i)T \) |
| 3 | \( 1 + (0.0348 + 0.999i)T \) |
| 5 | \( 1 + (0.438 + 0.898i)T \) |
| 7 | \( 1 + (0.990 - 0.139i)T \) |
| 13 | \( 1 + (-0.241 - 0.970i)T \) |
| 17 | \( 1 + (-0.241 + 0.970i)T \) |
| 19 | \( 1 + (0.0348 + 0.999i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.669 - 0.743i)T \) |
| 31 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.848 - 0.529i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.669 + 0.743i)T \) |
| 53 | \( 1 + (0.438 - 0.898i)T \) |
| 59 | \( 1 + (-0.882 + 0.469i)T \) |
| 61 | \( 1 + (-0.997 - 0.0697i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (0.559 - 0.829i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (-0.719 + 0.694i)T \) |
| 83 | \( 1 + (-0.241 + 0.970i)T \) |
| 89 | \( 1 + (0.173 + 0.984i)T \) |
| 97 | \( 1 + (0.913 - 0.406i)T \) |
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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
−24.01134168543430639684138471399, −23.161481029144678011757370126527, −21.92724250666742378898520035541, −21.282080976821051606217275473312, −20.240677839325832283303818045604, −19.87755592710516903918441372800, −18.606849281793747400800035733009, −18.022586387571187410703635576787, −17.17366195401349150504547935405, −15.92779456167484507492110325253, −14.41952488155091177824227719190, −14.03140276583171372425615552533, −13.12912372202455376989670900263, −12.23544540032829732728513981517, −11.67197981734263886358734793907, −10.70621426126808609309894977441, −9.16163567896597418239976650402, −8.75656621683758307415146645473, −7.322675801000135446202290373555, −6.16011103099231489853148188418, −5.093857665469148773750857838903, −4.44116034576176951737988346352, −2.642363763921997480300685273199, −1.87679443563101764366534756404, −0.869332920249461964820596250802,
2.29588320386611269839674952834, 3.506734149739600608727851695784, 4.300411631796742328620154997347, 5.552980435830977538725205762644, 6.006037255212673769122885720590, 7.56151666191321628489751757109, 8.178747084619858489343573451860, 9.43556951337049761349978723026, 10.45901071571233808578629181141, 11.235574233000893387594398238834, 12.405359935411620331216811824942, 13.75950677063874851601638589241, 14.3665001988014174154379704379, 15.09634163054946587149016021319, 15.652677805504898702785150933738, 16.980404106487632076394040390281, 17.48452081291183674662953930602, 18.28295526568964064541993342916, 19.76752017963301841623519056724, 20.97776345577032269218531820326, 21.377507481527413140315367563312, 22.32031256171022962705511593737, 22.84060291620200016599171801558, 23.848237974944593774741664354515, 24.863829284784606322717870162196