L(s) = 1 | + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.733 − 0.680i)5-s + (0.222 − 0.974i)8-s + (−0.955 − 0.294i)10-s + (−0.826 − 0.563i)11-s + (−0.826 − 0.563i)13-s + (−0.222 − 0.974i)16-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (−0.988 + 0.149i)20-s + (−0.988 − 0.149i)22-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.988 − 0.149i)26-s + ⋯ |
L(s) = 1 | + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.733 − 0.680i)5-s + (0.222 − 0.974i)8-s + (−0.955 − 0.294i)10-s + (−0.826 − 0.563i)11-s + (−0.826 − 0.563i)13-s + (−0.222 − 0.974i)16-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (−0.988 + 0.149i)20-s + (−0.988 − 0.149i)22-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.988 − 0.149i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3087489124 - 1.375279378i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3087489124 - 1.375279378i\) |
\(L(1)\) |
\(\approx\) |
\(1.060444819 - 0.7678082088i\) |
\(L(1)\) |
\(\approx\) |
\(1.060444819 - 0.7678082088i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.900 - 0.433i)T \) |
| 5 | \( 1 + (-0.733 - 0.680i)T \) |
| 11 | \( 1 + (-0.826 - 0.563i)T \) |
| 13 | \( 1 + (-0.826 - 0.563i)T \) |
| 17 | \( 1 + (-0.988 + 0.149i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.365 - 0.930i)T \) |
| 29 | \( 1 + (0.988 - 0.149i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.365 - 0.930i)T \) |
| 41 | \( 1 + (0.955 - 0.294i)T \) |
| 43 | \( 1 + (0.955 + 0.294i)T \) |
| 47 | \( 1 + (-0.900 + 0.433i)T \) |
| 53 | \( 1 + (-0.365 - 0.930i)T \) |
| 59 | \( 1 + (-0.222 - 0.974i)T \) |
| 61 | \( 1 + (-0.623 - 0.781i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (-0.826 + 0.563i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.826 - 0.563i)T \) |
| 89 | \( 1 + (0.0747 + 0.997i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−24.014657348491473750953833519499, −23.73289968750472815673222250230, −22.69789055647153033560649899245, −22.05988087938019789096442760256, −21.29915226169653124259054337317, −20.06458288655732786546271461868, −19.60134958895551122234482533196, −18.20535850353438345072937419017, −17.518092264755614611039272632467, −16.2517962380770252470515797237, −15.56354114367197812032204911, −14.94138215946122117500660974436, −14.000609867397611571056932738431, −13.13072341883703718333693168104, −12.098985225038888652070226181109, −11.41070080972349928299455204385, −10.48650806233287669418457820008, −9.1121054215237144492368377455, −7.725871422293715532353706038570, −7.26808123783545188416905451186, −6.3265071478813749961664381572, −4.98124660305006963932514457787, −4.30607393342085758848708202482, −3.05647684055077290133861913797, −2.25235784193731467776738006939,
0.55159520863740881381858723260, 2.150721952368702519182606096871, 3.2463922876116305245289851851, 4.31697399733701339551468888155, 5.11948577933421184234925817975, 6.06890551408956922373571554728, 7.400311123849923911907335666, 8.26342287991494170155886086463, 9.55237058885874438275256269753, 10.64534377069420167077081668611, 11.373558404370038030447736863597, 12.57228833648632988611715352616, 12.75928838521927594683591349508, 14.02147599304958033837446468462, 14.85750081828704610333180617127, 15.93119990493690431181248693993, 16.248962286789626775257425960798, 17.701847749207980246140376405072, 18.83651324340418634759091088326, 19.67066572142649074008268864717, 20.3231672115828291775142172993, 21.0761913657415175259478974151, 22.034253724860520770186845169442, 22.84434927371497790322549338775, 23.62360893348974132398343000026