L(s) = 1 | − 2-s + (0.5 + 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)15-s + 16-s − 17-s + (0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + ⋯ |
L(s) = 1 | − 2-s + (0.5 + 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s − 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)15-s + 16-s − 17-s + (0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2151064502 + 0.6247675271i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2151064502 + 0.6247675271i\) |
\(L(1)\) |
\(\approx\) |
\(0.6106830440 + 0.2467690916i\) |
\(L(1)\) |
\(\approx\) |
\(0.6106830440 + 0.2467690916i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−29.7903411028914696390112763083, −28.814425833175318185028905549657, −27.41229896914112040631111329430, −26.508471382197834857480262678548, −25.77801292715841176471599129472, −24.57681994928873895665000887610, −23.8411500089441045506220783710, −22.34841564424122836982332181334, −20.8533697452057705271929894364, −19.50839151767786844489816865994, −19.16481609085305524947768662745, −18.095849141108305689651192357119, −17.104191825600582518251050309269, −15.54362338851773342831342697365, −14.64510045165914288960962188284, −13.230659678736312239943180088536, −11.6585206186284410045455488870, −10.95739216662171378006943706476, −9.23450612820734506777682287273, −8.25372555161474450555614716225, −7.08426032140180207303887873495, −6.313492854207954929178026211649, −3.41308949452995160361720356217, −2.196413538767724232037338944393, −0.37603018401740660124128768566,
1.7917563806293439042133031726, 3.63674699196909575555596071052, 5.06462843012998299638659912496, 7.04673066591743698566387820733, 8.44874597364475934417901273958, 9.10131144960979223551275717546, 10.26854005333906741390694225760, 11.474081237383580897388252781960, 12.76957648650929498925668712439, 14.69006432289200646583458275363, 15.57112391775004540222998404146, 16.5484467386195881748916048507, 17.39203731196113808700126233578, 19.006914952341293141793632956018, 20.07076434014292331174251638188, 20.51450127820592646205641320957, 21.73310739751606688027673066332, 23.280933058461404572084325506388, 24.74749995410721174733429893181, 25.389369300101720680496680052385, 26.62510725000518041433122374662, 27.44942185686683267723872022608, 28.11674725912621303718839627161, 29.11386124236655447048042739017, 30.68505635012581826850719587784