L(s) = 1 | + (0.415 + 0.909i)3-s + (0.540 − 0.841i)7-s + (−0.654 + 0.755i)9-s + (−0.989 + 0.142i)11-s + (0.841 − 0.540i)13-s + (−0.281 − 0.959i)17-s + (−0.281 + 0.959i)19-s + (0.989 + 0.142i)21-s + (−0.959 − 0.281i)27-s + (−0.281 − 0.959i)29-s + (0.415 − 0.909i)31-s + (−0.540 − 0.841i)33-s + (0.654 − 0.755i)37-s + (0.841 + 0.540i)39-s + (0.654 + 0.755i)41-s + ⋯ |
L(s) = 1 | + (0.415 + 0.909i)3-s + (0.540 − 0.841i)7-s + (−0.654 + 0.755i)9-s + (−0.989 + 0.142i)11-s + (0.841 − 0.540i)13-s + (−0.281 − 0.959i)17-s + (−0.281 + 0.959i)19-s + (0.989 + 0.142i)21-s + (−0.959 − 0.281i)27-s + (−0.281 − 0.959i)29-s + (0.415 − 0.909i)31-s + (−0.540 − 0.841i)33-s + (0.654 − 0.755i)37-s + (0.841 + 0.540i)39-s + (0.654 + 0.755i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.728347250 - 0.1797658092i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.728347250 - 0.1797658092i\) |
\(L(1)\) |
\(\approx\) |
\(1.208936135 + 0.1257546644i\) |
\(L(1)\) |
\(\approx\) |
\(1.208936135 + 0.1257546644i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.415 + 0.909i)T \) |
| 7 | \( 1 + (0.540 - 0.841i)T \) |
| 11 | \( 1 + (-0.989 + 0.142i)T \) |
| 13 | \( 1 + (0.841 - 0.540i)T \) |
| 17 | \( 1 + (-0.281 - 0.959i)T \) |
| 19 | \( 1 + (-0.281 + 0.959i)T \) |
| 29 | \( 1 + (-0.281 - 0.959i)T \) |
| 31 | \( 1 + (0.415 - 0.909i)T \) |
| 37 | \( 1 + (0.654 - 0.755i)T \) |
| 41 | \( 1 + (0.654 + 0.755i)T \) |
| 43 | \( 1 + (0.415 + 0.909i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.841 + 0.540i)T \) |
| 59 | \( 1 + (-0.540 - 0.841i)T \) |
| 61 | \( 1 + (0.909 + 0.415i)T \) |
| 67 | \( 1 + (-0.142 + 0.989i)T \) |
| 71 | \( 1 + (0.142 - 0.989i)T \) |
| 73 | \( 1 + (-0.281 + 0.959i)T \) |
| 79 | \( 1 + (0.841 - 0.540i)T \) |
| 83 | \( 1 + (0.654 - 0.755i)T \) |
| 89 | \( 1 + (-0.415 - 0.909i)T \) |
| 97 | \( 1 + (0.755 - 0.654i)T \) |
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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
−20.11484991928951810024804921574, −19.217908880132020574600831177, −18.78046686082712673336691848201, −17.94379186061876878336360427558, −17.64760017140348982411910012653, −16.491558663601777672328690340807, −15.53679944751847245576348497370, −15.05330417603512616955320302490, −14.14311957750815765374353187181, −13.47597521530502357915054783579, −12.756941496981217506240184552356, −12.156387353624731282980180745437, −11.18346594691924824134779970830, −10.696801320224776669017279787123, −9.307117488292915451596353789948, −8.63256802525420805197936920689, −8.2119204900542301319305606371, −7.25367634865623962088441725438, −6.41611068785023587700254560156, −5.70085064466527172581822129618, −4.803084943272324787557292023648, −3.638793148303808685992572311158, −2.651078719794641751599591993529, −2.022132789686154434569818272404, −1.06087165636120999077511807092,
0.643633705959028940123116028470, 2.068336180313729567142276944739, 2.91213255630940408207000741089, 3.90735216332647834754629468347, 4.47081357168399718590842680824, 5.36762079194510082300498469458, 6.14274599029630080217828204125, 7.59165009897763476080862098055, 7.88863243025073473965157066167, 8.76370474525182341308577734904, 9.78069320356310840283413066420, 10.291882335695903452922336748153, 11.04703005702487089421765695047, 11.600644498624440424266364585318, 13.01979946821848532242313152049, 13.47544520789522634310351899789, 14.29064495509922189318697943607, 14.97311063226891524533932331493, 15.74611966508679228675041382458, 16.32310964298846734820131582541, 17.058617625417774447254408650206, 17.94896639549203452729838934960, 18.57431117088190506758561050064, 19.614994547943639891141098890476, 20.35105256314033170072381234176