L(s) = 1 | + (−0.866 + 0.5i)2-s − 3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.866 − 0.5i)6-s + i·8-s + 9-s + 10-s + i·11-s + (−0.5 + 0.866i)12-s + (0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)18-s + i·19-s + (−0.866 + 0.5i)20-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s − 3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.866 − 0.5i)6-s + i·8-s + 9-s + 10-s + i·11-s + (−0.5 + 0.866i)12-s + (0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)18-s + i·19-s + (−0.866 + 0.5i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2266624442 + 0.2507845390i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2266624442 + 0.2507845390i\) |
\(L(1)\) |
\(\approx\) |
\(0.4218083866 + 0.1453174363i\) |
\(L(1)\) |
\(\approx\) |
\(0.4218083866 + 0.1453174363i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + iT \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.866 - 0.5i)T \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 + (-0.866 - 0.5i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (-0.866 + 0.5i)T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−29.96169655554313894486809326661, −28.92346715321668857611293023134, −28.09948303933160854698951188237, −26.92394062999619330729668873761, −26.64713375963792878590727913269, −24.85645836427665988338630323981, −23.78950898923428797312363415376, −22.51478228525711011062262103607, −21.741633495532848223810305972857, −20.39379171665781765344994459187, −19.09015089855765445161530848157, −18.47643842854542243760688471791, −17.29807694329131897996252674535, −16.244239288986045683905766015586, −15.420000295157763319431309643534, −13.32054516818463415463708836739, −11.88920225293737048728704880434, −11.275047588119404572117759829212, −10.371984405160754425048854442563, −8.825591436746465926940905962078, −7.43682175531797197262170642761, −6.43053284555456787938630270453, −4.4365758447132686764854341688, −2.86984921752281693248961126814, −0.578629855156341805400157712072,
1.45925254485628420937393099144, 4.321359521966403997775401361251, 5.61047068534321134305033041001, 6.97166915078106274813127587019, 7.96883935175963794405629157002, 9.4428134797551405865883066328, 10.655339355305079885441167492363, 11.712760805632240393812291811967, 12.79670775685329336209916180804, 14.92472259554428430781780266526, 15.76025790855635159249012028453, 16.77551593156854733028793519644, 17.5749382772466743832783809090, 18.72526443240638184937308373414, 19.76503543844606018668732453252, 20.92284375396121975431570819497, 22.64773683119620452043398477877, 23.5035414871175391270176648982, 24.26724988576062458723429308266, 25.436654935142074783209571781701, 26.81354817346917782502365951476, 27.6454303889867777094480212182, 28.32612578532875601350475498785, 29.19549264412844883102296019209, 30.53574558943461611391007759688