L(s) = 1 | + (−0.948 + 0.316i)2-s + (0.987 − 0.160i)3-s + (0.799 − 0.600i)4-s + (−0.692 + 0.721i)5-s + (−0.885 + 0.464i)6-s + (−0.568 + 0.822i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (−0.948 − 0.316i)11-s + (0.692 − 0.721i)12-s + (−0.568 + 0.822i)15-s + (0.278 − 0.960i)16-s + (−0.996 − 0.0804i)17-s + (−0.799 + 0.600i)18-s + (0.5 − 0.866i)19-s + (−0.120 + 0.992i)20-s + ⋯ |
L(s) = 1 | + (−0.948 + 0.316i)2-s + (0.987 − 0.160i)3-s + (0.799 − 0.600i)4-s + (−0.692 + 0.721i)5-s + (−0.885 + 0.464i)6-s + (−0.568 + 0.822i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (−0.948 − 0.316i)11-s + (0.692 − 0.721i)12-s + (−0.568 + 0.822i)15-s + (0.278 − 0.960i)16-s + (−0.996 − 0.0804i)17-s + (−0.799 + 0.600i)18-s + (0.5 − 0.866i)19-s + (−0.120 + 0.992i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6520437519 - 0.4522471468i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6520437519 - 0.4522471468i\) |
\(L(1)\) |
\(\approx\) |
\(0.7640831615 + 0.002212430352i\) |
\(L(1)\) |
\(\approx\) |
\(0.7640831615 + 0.002212430352i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.948 + 0.316i)T \) |
| 3 | \( 1 + (0.987 - 0.160i)T \) |
| 5 | \( 1 + (-0.692 + 0.721i)T \) |
| 11 | \( 1 + (-0.948 - 0.316i)T \) |
| 17 | \( 1 + (-0.996 - 0.0804i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.748 - 0.663i)T \) |
| 31 | \( 1 + (0.845 + 0.534i)T \) |
| 37 | \( 1 + (0.0402 - 0.999i)T \) |
| 41 | \( 1 + (0.354 - 0.935i)T \) |
| 43 | \( 1 + (0.885 + 0.464i)T \) |
| 47 | \( 1 + (-0.799 - 0.600i)T \) |
| 53 | \( 1 + (-0.996 - 0.0804i)T \) |
| 59 | \( 1 + (-0.278 - 0.960i)T \) |
| 61 | \( 1 + (-0.996 + 0.0804i)T \) |
| 67 | \( 1 + (0.919 - 0.391i)T \) |
| 71 | \( 1 + (0.354 - 0.935i)T \) |
| 73 | \( 1 + (-0.948 - 0.316i)T \) |
| 79 | \( 1 + (0.799 + 0.600i)T \) |
| 83 | \( 1 + (0.354 + 0.935i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.970 - 0.239i)T \) |
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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
−20.87416979711724126996574417209, −20.40276211512076573673541403293, −20.02783381101865059175097023783, −19.03354480678045247597790861139, −18.57050453957886315742659116241, −17.67983869265025292157661157092, −16.61096625786476817067951264343, −15.970719172830752816800682074749, −15.45583886325690619402061397588, −14.608641526432214312049714817943, −13.324842050912249960643853073424, −12.76358795474177534921530205919, −11.972205163319267070217830344928, −10.96337106570559482604376277643, −10.158023214404461952530227831, −9.38729885994770487760874105905, −8.63240126570144059167867728697, −7.93231811142762051499627562971, −7.52360197312944484523403791996, −6.33527409504442805117629042145, −4.83407243740943967698754836482, −4.02932050306532166151154991406, −3.05541447204586086801843659273, −2.19372152607848386642809073070, −1.18672184181359775420014326827,
0.39684093019911942603010940132, 1.95573027278087479708221235328, 2.692626708666362995719956595070, 3.51560155303982806212412544886, 4.76638423436124297081528082445, 6.075718664516925822901979650259, 7.002903823859925593249243700005, 7.62493764763541141338994404969, 8.18022348131699439361263550045, 9.09508835967252713794156925041, 9.80724320057486495755820590632, 10.79246579881092369600008785125, 11.31448551883426313051018898514, 12.41897317045539937266313916691, 13.55713116920620150649268393032, 14.18279104864680135704006979469, 15.22091081713997991757822156547, 15.61749380553446647079223963394, 16.1075039135549191559672330075, 17.58729199436017473972064964663, 18.07034474361695918049960617518, 18.83978378885185708650920314574, 19.48824150183022678577343062557, 19.9423696308263863835986288646, 20.82910239854383057268207092266