L(s) = 1 | + (0.406 + 0.913i)2-s + (−0.743 + 0.669i)3-s + (−0.669 + 0.743i)4-s + (−0.913 − 0.406i)6-s + (−0.951 + 0.309i)7-s + (−0.951 − 0.309i)8-s + (0.104 − 0.994i)9-s − i·12-s + (0.994 + 0.104i)13-s + (−0.669 − 0.743i)14-s + (−0.104 − 0.994i)16-s + (0.994 − 0.104i)17-s + (0.951 − 0.309i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (0.913 − 0.406i)24-s + ⋯ |
L(s) = 1 | + (0.406 + 0.913i)2-s + (−0.743 + 0.669i)3-s + (−0.669 + 0.743i)4-s + (−0.913 − 0.406i)6-s + (−0.951 + 0.309i)7-s + (−0.951 − 0.309i)8-s + (0.104 − 0.994i)9-s − i·12-s + (0.994 + 0.104i)13-s + (−0.669 − 0.743i)14-s + (−0.104 − 0.994i)16-s + (0.994 − 0.104i)17-s + (0.951 − 0.309i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (0.913 − 0.406i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7102536620 + 0.9227369417i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7102536620 + 0.9227369417i\) |
\(L(1)\) |
\(\approx\) |
\(0.7192446297 + 0.5915472125i\) |
\(L(1)\) |
\(\approx\) |
\(0.7192446297 + 0.5915472125i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.406 + 0.913i)T \) |
| 3 | \( 1 + (-0.743 + 0.669i)T \) |
| 7 | \( 1 + (-0.951 + 0.309i)T \) |
| 13 | \( 1 + (0.994 + 0.104i)T \) |
| 17 | \( 1 + (0.994 - 0.104i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.669 - 0.743i)T \) |
| 31 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (0.951 - 0.309i)T \) |
| 41 | \( 1 + (-0.669 - 0.743i)T \) |
| 43 | \( 1 + (0.866 + 0.5i)T \) |
| 47 | \( 1 + (0.207 + 0.978i)T \) |
| 53 | \( 1 + (-0.994 - 0.104i)T \) |
| 59 | \( 1 + (0.978 + 0.207i)T \) |
| 61 | \( 1 + (-0.913 - 0.406i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (-0.104 - 0.994i)T \) |
| 73 | \( 1 + (-0.207 + 0.978i)T \) |
| 79 | \( 1 + (0.913 - 0.406i)T \) |
| 83 | \( 1 + (0.587 + 0.809i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.406 - 0.913i)T \) |
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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
−21.479671274964385837996030358051, −20.47571660833410734099230283278, −19.70102989099529264490663523082, −18.98713695952994003620767978211, −18.44455375068725945096189340906, −17.64185351321205757815710360019, −16.66203903480356996823378829560, −15.98136214950662234050351714744, −14.83774083784701552640979584425, −13.78200989783540657959527634143, −13.24690396748229891666133065530, −12.570906215565654507510877909238, −11.90355595682711501160290958363, −10.9470333630405702086552791099, −10.438491011068850088326743676834, −9.5019823292604170308580098246, −8.50846588918228100451979690476, −7.31699286336233934077208020568, −6.37238789579074393873609124399, −5.705894527812736741189709272735, −4.82196735116519506305942120037, −3.62288729129602082293398384417, −2.93220199377667582894382782351, −1.57210544055757608821456231207, −0.80988617717813426561254109584,
0.75176722485578107090580361407, 2.90219626265702682888243707332, 3.68904898908664108120213329926, 4.50738453747590806227764736223, 5.57102124137070103416771809055, 6.086998539633243959917474284648, 6.80429375931246731999397173041, 7.89566606291683914866488665727, 9.02760282604812027779882299141, 9.519891029575613770498635569643, 10.54487563386579521652423627964, 11.54779333389141087399212743353, 12.43248950144389577050083263835, 13.013494679998397592886141251142, 14.026142714097040167806448139333, 14.94943728559969544508071384142, 15.61798289536368760481948807498, 16.32580206989794941633288171707, 16.71245888591867047744132244554, 17.66370652648137829818913258049, 18.468684150232292870180329725247, 19.166570916705956477639600058566, 20.69184253724164386391366011665, 21.10785045386938333677458794495, 22.09481870198138201335688068042