L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s − 6-s − i·8-s + (0.5 − 0.866i)9-s + (0.866 + 0.5i)12-s − i·13-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (−0.5 + 0.866i)19-s + (−0.866 − 0.5i)23-s + (−0.5 − 0.866i)24-s + (−0.5 + 0.866i)26-s − i·27-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s − 6-s − i·8-s + (0.5 − 0.866i)9-s + (0.866 + 0.5i)12-s − i·13-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (−0.5 + 0.866i)19-s + (−0.866 − 0.5i)23-s + (−0.5 − 0.866i)24-s + (−0.5 + 0.866i)26-s − i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0932 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0932 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7737107342 - 0.8495312596i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7737107342 - 0.8495312596i\) |
\(L(1)\) |
\(\approx\) |
\(0.8624655288 - 0.4471924152i\) |
\(L(1)\) |
\(\approx\) |
\(0.8624655288 - 0.4471924152i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 - iT \) |
| 17 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.866 - 0.5i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 - iT \) |
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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
−25.10553355345448255527357780805, −23.98741006268091568411136284796, −23.39350867882734157971950017653, −21.79531053045546379997845326950, −21.2247817549498210589407018522, −20.05621575747597841540106878474, −19.49735882735277245859943261175, −18.721983851723395005533432815568, −17.703849534973629738803636477697, −16.61161516868465430789344240429, −16.01962457330505330166859995497, −15.044845849881353177862908802636, −14.35069275652124637227650819223, −13.50090605493684398318507040379, −11.99449153682271685523314514948, −10.82323365184995043726367865267, −10.001613789529682293279686625048, −9.15716574814700171781250399842, −8.406387534406179484725269524188, −7.48902773392726884859417353383, −6.48545830737205112933298816417, −5.18983693091600814479398359852, −4.03271143546374597822973967740, −2.61801359495751077843393731441, −1.52942682820151438858880222217,
0.8752484868751711554227626938, 2.13317136083725341833756906986, 3.06317421516655570758791645201, 4.0507120261266061564312095141, 5.93809749852420089688074562547, 7.142508952298919249641702437017, 8.00685905952676955004344291079, 8.582169863957092078529858646419, 9.81106870214376878431185186465, 10.35211251225032595073101134731, 11.81001971009445371853214104421, 12.49165191810880433231381268363, 13.37992447165177978727621876919, 14.486016496686996957276142009775, 15.46004779201657798761752647000, 16.472064897413647220457035677022, 17.49868824349110746204448503834, 18.445040547745417370414361092718, 18.87586076962606295684106656544, 20.01115812792988926333523542239, 20.4443182736394052580974194264, 21.29206673565008106546243472645, 22.374719639823191612905096128355, 23.56046491700381696952429967328, 24.64105833145871090825696265299