L(s) = 1 | + (−0.406 − 0.913i)2-s + (0.207 + 0.978i)3-s + (−0.669 + 0.743i)4-s + (0.809 − 0.587i)6-s + (0.951 + 0.309i)8-s + (−0.913 + 0.406i)9-s + (−0.866 − 0.5i)12-s + (0.587 − 0.809i)13-s + (−0.104 − 0.994i)16-s + (0.406 − 0.913i)17-s + (0.743 + 0.669i)18-s + (0.669 + 0.743i)19-s + (0.866 + 0.5i)23-s + (−0.104 + 0.994i)24-s + (−0.978 − 0.207i)26-s + (−0.587 − 0.809i)27-s + ⋯ |
L(s) = 1 | + (−0.406 − 0.913i)2-s + (0.207 + 0.978i)3-s + (−0.669 + 0.743i)4-s + (0.809 − 0.587i)6-s + (0.951 + 0.309i)8-s + (−0.913 + 0.406i)9-s + (−0.866 − 0.5i)12-s + (0.587 − 0.809i)13-s + (−0.104 − 0.994i)16-s + (0.406 − 0.913i)17-s + (0.743 + 0.669i)18-s + (0.669 + 0.743i)19-s + (0.866 + 0.5i)23-s + (−0.104 + 0.994i)24-s + (−0.978 − 0.207i)26-s + (−0.587 − 0.809i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.032960835 + 0.1898321703i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.032960835 + 0.1898321703i\) |
\(L(1)\) |
\(\approx\) |
\(0.9107928460 + 0.005450793329i\) |
\(L(1)\) |
\(\approx\) |
\(0.9107928460 + 0.005450793329i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.406 - 0.913i)T \) |
| 3 | \( 1 + (0.207 + 0.978i)T \) |
| 13 | \( 1 + (0.587 - 0.809i)T \) |
| 17 | \( 1 + (0.406 - 0.913i)T \) |
| 19 | \( 1 + (0.669 + 0.743i)T \) |
| 23 | \( 1 + (0.866 + 0.5i)T \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.104 + 0.994i)T \) |
| 37 | \( 1 + (0.207 - 0.978i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (-0.743 + 0.669i)T \) |
| 53 | \( 1 + (0.994 + 0.104i)T \) |
| 59 | \( 1 + (-0.669 + 0.743i)T \) |
| 61 | \( 1 + (0.104 + 0.994i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.743 + 0.669i)T \) |
| 79 | \( 1 + (0.913 - 0.406i)T \) |
| 83 | \( 1 + (-0.587 - 0.809i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.587 - 0.809i)T \) |
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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
−24.428045367685063186921129925831, −23.8245522095734212463063082618, −23.117812923217231552628547638001, −22.153410708874194401898876862804, −20.815269681226322949243040631084, −19.73293740124028093716186092085, −18.89353182922214307840998453153, −18.43063022981135057073244836639, −17.30350964733737226221773964805, −16.79660850794113791459747956825, −15.5407298029526691227741450711, −14.7295468467316824150979817638, −13.74024176504416894412341015056, −13.207617588622536593167376425431, −11.94968898357549719816185081333, −10.87984296340827472665911548312, −9.567995173092198629896365047333, −8.6709666352760912360533977895, −7.91549850694152419467894043488, −6.887227388404555594354948487331, −6.25100225945387925659386965797, −5.14410336364837131989052988754, −3.73096280665698585628706304059, −2.08633496049668672949554265290, −0.86505501697994332052928122484,
1.21831373272111885548663923887, 2.91359649623246622977031671912, 3.42689385292641371516440513070, 4.69901836612417238752438830843, 5.573629281557405439812272420123, 7.43162678662095501514999954332, 8.42113499707904835025162376016, 9.281961373075024680086074970873, 10.08493204956298368569071832025, 10.89086969623664953935941266183, 11.69406555197784824318625447364, 12.80814755557759838372607768093, 13.83506711571888152255006810198, 14.68384198053054828714877809646, 15.98335979586244674448575235776, 16.532439823641384350979785354432, 17.72014152669915853700263734137, 18.40791011422129188630934588758, 19.630874923913104997848855393975, 20.21893883034525841085041753598, 21.06829518162548574272895330007, 21.60476603298266707910537692485, 22.82321085974703264318939045654, 23.03878822906700273260896008708, 24.972152314650934713652082489040