L(s) = 1 | + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.994 + 0.104i)5-s + (−0.587 − 0.809i)8-s + (0.104 + 0.994i)10-s + (0.777 + 0.629i)11-s + (−0.453 − 0.891i)13-s + (0.669 − 0.743i)16-s + (−0.629 + 0.777i)17-s + (0.998 − 0.0523i)19-s + (−0.951 + 0.309i)20-s + (−0.453 + 0.891i)22-s + (−0.978 + 0.207i)23-s + (0.978 + 0.207i)25-s + (0.777 − 0.629i)26-s + ⋯ |
L(s) = 1 | + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.994 + 0.104i)5-s + (−0.587 − 0.809i)8-s + (0.104 + 0.994i)10-s + (0.777 + 0.629i)11-s + (−0.453 − 0.891i)13-s + (0.669 − 0.743i)16-s + (−0.629 + 0.777i)17-s + (0.998 − 0.0523i)19-s + (−0.951 + 0.309i)20-s + (−0.453 + 0.891i)22-s + (−0.978 + 0.207i)23-s + (0.978 + 0.207i)25-s + (0.777 − 0.629i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2402504059 + 1.983239482i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2402504059 + 1.983239482i\) |
\(L(1)\) |
\(\approx\) |
\(0.9687441024 + 0.7484347618i\) |
\(L(1)\) |
\(\approx\) |
\(0.9687441024 + 0.7484347618i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.207 + 0.978i)T \) |
| 5 | \( 1 + (0.994 + 0.104i)T \) |
| 11 | \( 1 + (0.777 + 0.629i)T \) |
| 13 | \( 1 + (-0.453 - 0.891i)T \) |
| 17 | \( 1 + (-0.629 + 0.777i)T \) |
| 19 | \( 1 + (0.998 - 0.0523i)T \) |
| 23 | \( 1 + (-0.978 + 0.207i)T \) |
| 29 | \( 1 + (-0.987 + 0.156i)T \) |
| 31 | \( 1 + (-0.104 - 0.994i)T \) |
| 37 | \( 1 + (-0.104 + 0.994i)T \) |
| 43 | \( 1 + (0.951 + 0.309i)T \) |
| 47 | \( 1 + (0.838 + 0.544i)T \) |
| 53 | \( 1 + (-0.358 + 0.933i)T \) |
| 59 | \( 1 + (0.669 + 0.743i)T \) |
| 61 | \( 1 + (-0.743 - 0.669i)T \) |
| 67 | \( 1 + (-0.933 - 0.358i)T \) |
| 71 | \( 1 + (-0.156 + 0.987i)T \) |
| 73 | \( 1 + (0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.965 - 0.258i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.0523 + 0.998i)T \) |
| 97 | \( 1 + (0.156 + 0.987i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.50333792058093309467914825640, −20.76730135787071918379822429111, −19.99820204752299408315564487577, −19.24775279607964409552328359294, −18.32465440336402544387907584317, −17.76681287511713019592922439926, −16.85534830893800289645649107350, −16.02226368286928797757203810525, −14.565170546213450033840632880467, −14.02170143319745886980369912751, −13.537719849411726054747751423517, −12.454244558847023030775029214365, −11.73038446411653916684534269281, −10.95966358447139155218891926423, −9.96807409053981232205215107241, −9.24810471900876648882446331597, −8.78640633377064536592277373984, −7.25779012639723559361677768454, −6.15521509774798140970288907075, −5.3684758354388598592150871119, −4.421961297221828890077358512078, −3.42871278496386214363399606666, −2.32483186464691324776928157971, −1.5954896941571207043380032638, −0.41814765887809423453842402547,
1.164991677478349384578328179201, 2.44337435404854421473221636691, 3.68410631483293984019850690447, 4.65760231297660252932991766787, 5.66032212624575903488846883558, 6.20134110710375760725034766853, 7.21082162407438863437381118542, 7.95682726239010294263826611855, 9.161132825624276977105378670792, 9.63237284057850455061828021930, 10.552104175938757094362501444739, 11.924547431835485628836811693811, 12.7904994707352317060285213660, 13.478425528963613557993220558666, 14.2919079825865613192470265503, 14.97119369515358020421545590384, 15.69909088752470773949494718742, 16.83163898856496195788630807062, 17.36558385934340718859841020044, 17.92682278602995697863204488699, 18.74773935444506168532322670723, 19.96743213889221921801768942888, 20.67842905222808777537308036255, 21.95433502791511040369469321974, 22.17956314505252871120170391551