L(s) = 1 | + (0.900 + 0.433i)3-s + (−0.222 − 0.974i)5-s + (0.900 + 0.433i)7-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + (0.623 − 0.781i)13-s + (0.222 − 0.974i)15-s + 17-s + (0.900 − 0.433i)19-s + (0.623 + 0.781i)21-s + (0.222 − 0.974i)23-s + (−0.900 + 0.433i)25-s + (0.222 + 0.974i)27-s + (0.222 + 0.974i)31-s + (−0.900 + 0.433i)33-s + ⋯ |
L(s) = 1 | + (0.900 + 0.433i)3-s + (−0.222 − 0.974i)5-s + (0.900 + 0.433i)7-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + (0.623 − 0.781i)13-s + (0.222 − 0.974i)15-s + 17-s + (0.900 − 0.433i)19-s + (0.623 + 0.781i)21-s + (0.222 − 0.974i)23-s + (−0.900 + 0.433i)25-s + (0.222 + 0.974i)27-s + (0.222 + 0.974i)31-s + (−0.900 + 0.433i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.605722684 + 0.1788322563i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.605722684 + 0.1788322563i\) |
\(L(1)\) |
\(\approx\) |
\(1.606695673 + 0.07978076503i\) |
\(L(1)\) |
\(\approx\) |
\(1.606695673 + 0.07978076503i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + (0.900 + 0.433i)T \) |
| 5 | \( 1 + (-0.222 - 0.974i)T \) |
| 7 | \( 1 + (0.900 + 0.433i)T \) |
| 11 | \( 1 + (-0.623 + 0.781i)T \) |
| 13 | \( 1 + (0.623 - 0.781i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (0.900 - 0.433i)T \) |
| 23 | \( 1 + (0.222 - 0.974i)T \) |
| 31 | \( 1 + (0.222 + 0.974i)T \) |
| 37 | \( 1 + (0.623 + 0.781i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (0.222 - 0.974i)T \) |
| 47 | \( 1 + (-0.623 + 0.781i)T \) |
| 53 | \( 1 + (-0.222 - 0.974i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (-0.900 - 0.433i)T \) |
| 67 | \( 1 + (-0.623 - 0.781i)T \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (-0.222 + 0.974i)T \) |
| 79 | \( 1 + (-0.623 - 0.781i)T \) |
| 83 | \( 1 + (0.900 - 0.433i)T \) |
| 89 | \( 1 + (-0.222 - 0.974i)T \) |
| 97 | \( 1 + (-0.900 + 0.433i)T \) |
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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
−29.398959414991892374681963206256, −27.71073841277574817583524892652, −26.606952237124626767949868296556, −26.19161001431328037167145486013, −24.93855491081815693757611468656, −23.85662042841250140452397258320, −23.16027043607564266345879270347, −21.468329254215089501368918775675, −20.86217559669217464737434054870, −19.53417382673814358877038916668, −18.633798598996926263906967483414, −17.97743766313745069675738031333, −16.310365561103356939558371959758, −15.05004637278292052772067766965, −14.1360001878777962196985769402, −13.52238331462595726638897456031, −11.81128026893615998966515845990, −10.83812106040269237587151758073, −9.48148435598963105638437414501, −7.97320704499929586777525812216, −7.457372843428398700510945476387, −5.93497380537523229369833880916, −3.97106430424402803736704815534, −2.894113142654038610669758214010, −1.34183773188298034020538857351,
1.36220418321631452417565698090, 2.93396509495398010884815343621, 4.52297500713767677812832313131, 5.34134751291679794546364707188, 7.668467476676202530479186091981, 8.33334314389066684851669253390, 9.420787397744016831169448383900, 10.64421336156950651957617007775, 12.163098610106171220559300819485, 13.159437891699457096910647440939, 14.396596150297252728785922608722, 15.41557845129034589277291575647, 16.19729883744089880569359368444, 17.6483857037064474281564180164, 18.72842540676859277404234868985, 20.16169471870365094238573260634, 20.65133624022906279534696198515, 21.45326280440632041082289558671, 22.95371278520664817706521574511, 24.18818103789480218104094982351, 24.99644793699384298430595799763, 25.846936440802815714041584887987, 27.17536189674638466051355360960, 27.85026318392769533059726166878, 28.712123682058487098620191746760