L(s) = 1 | + (0.876 − 0.481i)2-s + (0.535 − 0.844i)4-s + (−0.929 − 0.368i)5-s + (0.0627 − 0.998i)8-s + (0.728 − 0.684i)9-s + (−0.992 + 0.125i)10-s + (−0.929 + 0.872i)13-s + (−0.425 − 0.904i)16-s + (0.781 + 1.23i)17-s + (0.309 − 0.951i)18-s + (−0.809 + 0.587i)20-s + (0.728 + 0.684i)25-s + (−0.393 + 1.21i)26-s + (−0.824 − 0.211i)29-s + (−0.809 − 0.587i)32-s + ⋯ |
L(s) = 1 | + (0.876 − 0.481i)2-s + (0.535 − 0.844i)4-s + (−0.929 − 0.368i)5-s + (0.0627 − 0.998i)8-s + (0.728 − 0.684i)9-s + (−0.992 + 0.125i)10-s + (−0.929 + 0.872i)13-s + (−0.425 − 0.904i)16-s + (0.781 + 1.23i)17-s + (0.309 − 0.951i)18-s + (−0.809 + 0.587i)20-s + (0.728 + 0.684i)25-s + (−0.393 + 1.21i)26-s + (−0.824 − 0.211i)29-s + (−0.809 − 0.587i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.256453492\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.256453492\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.876 + 0.481i)T \) |
| 5 | \( 1 + (0.929 + 0.368i)T \) |
good | 3 | \( 1 + (-0.728 + 0.684i)T^{2} \) |
| 7 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 11 | \( 1 + (-0.535 + 0.844i)T^{2} \) |
| 13 | \( 1 + (0.929 - 0.872i)T + (0.0627 - 0.998i)T^{2} \) |
| 17 | \( 1 + (-0.781 - 1.23i)T + (-0.425 + 0.904i)T^{2} \) |
| 19 | \( 1 + (-0.728 - 0.684i)T^{2} \) |
| 23 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 29 | \( 1 + (0.824 + 0.211i)T + (0.876 + 0.481i)T^{2} \) |
| 31 | \( 1 + (0.425 - 0.904i)T^{2} \) |
| 37 | \( 1 + (-0.844 - 1.79i)T + (-0.637 + 0.770i)T^{2} \) |
| 41 | \( 1 + (1.06 - 0.134i)T + (0.968 - 0.248i)T^{2} \) |
| 43 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 47 | \( 1 + (0.992 - 0.125i)T^{2} \) |
| 53 | \( 1 + (1.23 + 1.49i)T + (-0.187 + 0.982i)T^{2} \) |
| 59 | \( 1 + (0.929 + 0.368i)T^{2} \) |
| 61 | \( 1 + (-1.84 - 0.233i)T + (0.968 + 0.248i)T^{2} \) |
| 67 | \( 1 + (-0.876 + 0.481i)T^{2} \) |
| 71 | \( 1 + (0.992 - 0.125i)T^{2} \) |
| 73 | \( 1 + (0.0235 + 0.123i)T + (-0.929 + 0.368i)T^{2} \) |
| 79 | \( 1 + (-0.728 + 0.684i)T^{2} \) |
| 83 | \( 1 + (-0.728 - 0.684i)T^{2} \) |
| 89 | \( 1 + (0.328 + 1.72i)T + (-0.929 + 0.368i)T^{2} \) |
| 97 | \( 1 + (0.362 + 0.0931i)T + (0.876 + 0.481i)T^{2} \) |
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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
−11.36290912119757999873271096769, −10.12184533574380495199770532954, −9.531035312622849805943993665545, −8.207795951932608861139227572163, −7.15720691335443332427059523582, −6.32889833170827018544446280104, −4.99092810148779155050946933560, −4.17317439495349933110254023257, −3.32887906439124947990127774348, −1.57786285725741538891877219249,
2.55255749585969842413646321168, 3.63026029367117938613158663860, 4.73262194946583383751823982928, 5.49215421847550422737571587987, 6.97097078066861598330095154218, 7.50963865539263680479678297604, 8.119436186935653428874083001686, 9.598372557249459331307448470336, 10.68463664629068779456110545823, 11.44467010617493900612808866396