L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s − 7-s + (−0.5 + 0.866i)9-s − 11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s + (−0.5 + 0.866i)29-s − 31-s + (−0.5 − 0.866i)33-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s − 7-s + (−0.5 + 0.866i)9-s − 11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s + (−0.5 + 0.866i)29-s − 31-s + (−0.5 − 0.866i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01752593053 + 0.1640539865i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01752593053 + 0.1640539865i\) |
\(L(1)\) |
\(\approx\) |
\(0.7044654399 + 0.1522540874i\) |
\(L(1)\) |
\(\approx\) |
\(0.7044654399 + 0.1522540874i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 - T \) |
| 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 | \( 1 + (0.5 + 0.866i)T \) |
| 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 | \( 1 + (-0.5 + 0.866i)T \) |
| 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
−30.49561981883184603385189571573, −29.56474150393750307728901488288, −28.66446321631262261226512418285, −26.97511855967602744670523203280, −26.049711762892050243587646552997, −25.307015595357778823462559361096, −23.861503345486333069016912416223, −23.06012294622356315460127874726, −21.9496540093739727883493167696, −20.2563162676432451720506971968, −19.349384456039528711077049851, −18.57988319194804094539414174974, −17.434131968122452183470968441924, −15.649158186390045337469313224577, −14.837162749267407710593636037118, −13.33921016024889981124715274568, −12.601719180397152173738229483540, −11.067960010178575064774801349491, −9.72344693831655193184406107201, −8.063174924982435991636341867981, −7.16158019752255778415155275196, −5.92246605051743434104751043150, −3.51770637190867887552957505464, −2.48626842802536689974483806011, −0.0702571172676305539353107504,
2.6770527478635420309532889247, 4.15214129033572408423262913625, 5.27780204062885009587922792953, 7.299543995787600802583119062176, 8.78600284889014487122995366433, 9.5788800467567226097097136179, 10.97035817670436579553586496021, 12.5094148464740004732470535683, 13.59690192595256285060133655279, 15.08952078175208113552367537957, 16.16228342721587607669769367547, 16.65452188294457496574819737460, 18.67034205808104969224464642991, 19.82632296161231857756914367241, 20.57751367733846754172852654412, 21.72557519187364443226606572055, 22.86384921416568297837926047554, 24.09814777904529875543643579986, 25.33727098838541836669239728492, 26.37682429687118586636232301053, 27.207491688666355981305211450400, 28.49218280297833190861568541714, 29.11984710604337764867700044421, 31.14571242120998598609652371868, 31.61521664767944375999915772328