L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + i·5-s + (−0.866 − 0.5i)7-s − i·8-s + (0.5 + 0.866i)10-s + (−0.866 + 0.5i)11-s − 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.866 + 0.5i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s − 25-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + i·5-s + (−0.866 − 0.5i)7-s − i·8-s + (0.5 + 0.866i)10-s + (−0.866 + 0.5i)11-s − 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.866 + 0.5i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s − 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.091680137 - 0.3152253091i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.091680137 - 0.3152253091i\) |
\(L(1)\) |
\(\approx\) |
\(1.309352061 - 0.2938183146i\) |
\(L(1)\) |
\(\approx\) |
\(1.309352061 - 0.2938183146i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.866 + 0.5i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 + (0.866 - 0.5i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (-0.866 - 0.5i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + T \) |
| 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
−35.24866494393519098847528364046, −34.09009108560918416031322091800, −32.73349609538945884017253872803, −31.917297154864845213487568002821, −31.10572539594234415118851424323, −29.3747347001883007877640848831, −28.56323721981635983341897288854, −26.70910401010646630334960340647, −25.41511926162147299081067111569, −24.465319051404606752004739202645, −23.37275821536238455608460168965, −22.07098332404063054473224248886, −20.96671903039801811739904519355, −19.7159259103815814671214822799, −17.79668449184314334782532395118, −16.15001454852903613804622847343, −15.74248807801981662824813660975, −13.72838440997076801374861968415, −12.84783644830002292148204968312, −11.60434732444579678329844912919, −9.32609519612752346890052481788, −7.82722605729856712753648415116, −6.03952750272900733390850490632, −4.81661252838882842811719405400, −2.932167245890278770202944209418,
2.550655005150280048324763697161, 3.965856835165729107788502321, 5.96716075357492301283651387094, 7.25208196541185733982899051650, 9.96135721152985103957219737700, 10.80338999148904019345750463890, 12.46203608168053526366124982173, 13.64979701360969999738762670439, 14.89578210679081190352577426472, 16.099699350134207202116911708833, 18.19218586998832770660884922619, 19.38671812014595237895865606321, 20.54007140588074506085714090267, 22.0337716991536068719830566479, 22.81735441628369306586682238299, 23.875321898851921749626352460092, 25.571480123759350791174828221449, 26.64520102730232883890401700039, 28.45642092789519855145870402041, 29.39259922889505116447657922061, 30.476866884860633835685338017965, 31.41517711643276767706869314577, 32.8067485055861462375206753926, 33.641631781691073516687761468761, 34.94840091752097655032284348874