L(s) = 1 | + (0.396 + 0.918i)2-s + (−0.686 + 0.727i)4-s + (−0.0581 + 0.998i)5-s + (0.973 − 0.230i)7-s + (−0.939 − 0.342i)8-s + (−0.939 + 0.342i)10-s + (−0.835 + 0.549i)11-s + (−0.993 + 0.116i)13-s + (0.597 + 0.802i)14-s + (−0.0581 − 0.998i)16-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.686 − 0.727i)20-s + (−0.835 − 0.549i)22-s + (0.973 + 0.230i)23-s + ⋯ |
L(s) = 1 | + (0.396 + 0.918i)2-s + (−0.686 + 0.727i)4-s + (−0.0581 + 0.998i)5-s + (0.973 − 0.230i)7-s + (−0.939 − 0.342i)8-s + (−0.939 + 0.342i)10-s + (−0.835 + 0.549i)11-s + (−0.993 + 0.116i)13-s + (0.597 + 0.802i)14-s + (−0.0581 − 0.998i)16-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.686 − 0.727i)20-s + (−0.835 − 0.549i)22-s + (0.973 + 0.230i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5805677255 + 0.9211334058i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5805677255 + 0.9211334058i\) |
\(L(1)\) |
\(\approx\) |
\(0.8926723115 + 0.7319929517i\) |
\(L(1)\) |
\(\approx\) |
\(0.8926723115 + 0.7319929517i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (0.396 + 0.918i)T \) |
| 5 | \( 1 + (-0.0581 + 0.998i)T \) |
| 7 | \( 1 + (0.973 - 0.230i)T \) |
| 11 | \( 1 + (-0.835 + 0.549i)T \) |
| 13 | \( 1 + (-0.993 + 0.116i)T \) |
| 17 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (0.766 - 0.642i)T \) |
| 23 | \( 1 + (0.973 + 0.230i)T \) |
| 29 | \( 1 + (0.597 - 0.802i)T \) |
| 31 | \( 1 + (-0.286 - 0.957i)T \) |
| 37 | \( 1 + (0.173 - 0.984i)T \) |
| 41 | \( 1 + (0.396 - 0.918i)T \) |
| 43 | \( 1 + (0.893 + 0.448i)T \) |
| 47 | \( 1 + (-0.286 + 0.957i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.835 - 0.549i)T \) |
| 61 | \( 1 + (-0.686 - 0.727i)T \) |
| 67 | \( 1 + (0.597 + 0.802i)T \) |
| 71 | \( 1 + (-0.939 + 0.342i)T \) |
| 73 | \( 1 + (-0.939 - 0.342i)T \) |
| 79 | \( 1 + (0.396 + 0.918i)T \) |
| 83 | \( 1 + (0.396 + 0.918i)T \) |
| 89 | \( 1 + (-0.939 - 0.342i)T \) |
| 97 | \( 1 + (-0.0581 - 0.998i)T \) |
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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
−30.860557697265767055381023345811, −29.37636447890178029198875258853, −28.82029616803557596077859696109, −27.50470127721742863225583440292, −27.00931690687988539602988849981, −24.925658299454634550166162502302, −24.10200418144850302103961218607, −23.14681543960622161365755425551, −21.66125269361211144000942772639, −20.92982157560949932083919966266, −20.091025804704880097734999405928, −18.79224700437183898873861161239, −17.73989043141732280797792133871, −16.33510027859512951277543783493, −14.831772070574937833709673213917, −13.753104740832762801251271961775, −12.50980441473130305683360228944, −11.70708652020958162340456988341, −10.36353543065501368868515054165, −9.060642767295216046412595967785, −7.87902985260329981942691539816, −5.37187700050843853542629625944, −4.826828945625309867916295877378, −2.95926889257868604527613272598, −1.270726801549941400468941468914,
2.71521232458012014816122388264, 4.40426135896923152165104539105, 5.63248459346330518422203139038, 7.26875446176399156475584938155, 7.80136252249749110460619682494, 9.63605251108097479916600461823, 11.07908240157990253125123494155, 12.476838208515292456804713355420, 13.915336310002339488803776517024, 14.75923869356930386827057302162, 15.57138249958401774187701377616, 17.19882557004835888362721675830, 17.893299812440918145257983975769, 19.084270110311622593594586770833, 20.8760522349453240314670263372, 21.832733022559817100432300079102, 22.94890467788409386017591278041, 23.79061620509221527870549591945, 24.85477770218605740755474225720, 26.10626023665248757243986982186, 26.73147400794318134797634915840, 27.81135626460366291094329222837, 29.53098147443102754339180508817, 30.69211531041409843287706384201, 31.16440001504374770668781967300