L(s) = 1 | + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)5-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + 11-s + (0.309 + 0.951i)13-s + (0.309 − 0.951i)15-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.309 + 0.951i)23-s + (0.309 − 0.951i)25-s + (0.309 + 0.951i)27-s + (−0.309 + 0.951i)29-s + (0.309 − 0.951i)31-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)5-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + 11-s + (0.309 + 0.951i)13-s + (0.309 − 0.951i)15-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.309 + 0.951i)23-s + (0.309 − 0.951i)25-s + (0.309 + 0.951i)27-s + (−0.309 + 0.951i)29-s + (0.309 − 0.951i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8894914746 + 0.2868981250i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8894914746 + 0.2868981250i\) |
\(L(1)\) |
\(\approx\) |
\(0.6937514480 + 0.2293184503i\) |
\(L(1)\) |
\(\approx\) |
\(0.6937514480 + 0.2293184503i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 \) |
good | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (0.309 + 0.951i)T \) |
| 17 | \( 1 + (0.809 + 0.587i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
| 23 | \( 1 + (-0.309 + 0.951i)T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (0.309 - 0.951i)T \) |
| 37 | \( 1 + (0.309 - 0.951i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + (0.809 + 0.587i)T \) |
| 47 | \( 1 + (0.809 - 0.587i)T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (0.809 - 0.587i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.309 - 0.951i)T \) |
| 71 | \( 1 + (-0.309 - 0.951i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.309 - 0.951i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + (-0.809 - 0.587i)T \) |
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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
−18.99708322938299871864367963000, −18.45351312287337736997055513386, −17.329914126034781520598262309842, −16.95847499169205829921511828837, −16.26427610232032550331821752421, −15.84371702348110101232643950464, −14.80332984828787262291041768296, −13.926399009912598295784225734895, −13.1776914799835752026254096439, −12.51105191281049277002075651782, −12.02766525364184939371571571863, −11.43366598722957961571841773379, −10.36537818774907371751692461762, −10.02618748433437328937497120315, −8.80736637952100183777719492672, −8.095786483789795346554712237049, −7.38997090944307390204707356285, −6.69627591574432803399917904221, −5.9504722819391260811020263444, −5.21777239935833385596941958744, −4.182906497766343121599541505594, −3.6925720535463804803772158042, −2.58557356085121207523315658735, −1.14005259866457793817740555364, −0.7939551554243818428588192749,
0.52105800838200178718191707918, 1.805513808411222852105804380, 3.04500399944254506372851574467, 3.82031780211840034870822357187, 4.17996090044957484380514576086, 5.3673555594759948789437910973, 6.11698105164450706311995692249, 6.721563004583644934027331113604, 7.32695245694288757391196234375, 8.579340586302976358937334145740, 9.25118830693584930888270087483, 9.84149350099266347445291099175, 10.762377365046885829321410030845, 11.40614450502163126301066574103, 11.9265127394235668315598420627, 12.482840774390330592076321612072, 13.47643162543725825904146990513, 14.52840145218262993417188807373, 15.04103104666185906186510097914, 15.68566827926953577252595449069, 16.37369837205508512922012684044, 16.82109784171063741184481817064, 17.73152797032414647100202087121, 18.46088272840455003609343935620, 19.25120652994209923342629955405