L(s) = 1 | + (−0.642 − 0.766i)5-s + (−0.642 + 0.766i)11-s + (−0.342 + 0.939i)13-s + 17-s − i·19-s + (−0.939 − 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.342 + 0.939i)29-s + (−0.173 − 0.984i)31-s + (0.866 + 0.5i)37-s + (0.939 + 0.342i)41-s + (−0.984 − 0.173i)43-s + (0.173 − 0.984i)47-s + (−0.866 − 0.5i)53-s + 55-s + ⋯ |
L(s) = 1 | + (−0.642 − 0.766i)5-s + (−0.642 + 0.766i)11-s + (−0.342 + 0.939i)13-s + 17-s − i·19-s + (−0.939 − 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.342 + 0.939i)29-s + (−0.173 − 0.984i)31-s + (0.866 + 0.5i)37-s + (0.939 + 0.342i)41-s + (−0.984 − 0.173i)43-s + (0.173 − 0.984i)47-s + (−0.866 − 0.5i)53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04753453653 - 0.2907363365i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04753453653 - 0.2907363365i\) |
\(L(1)\) |
\(\approx\) |
\(0.7678822623 - 0.08037093440i\) |
\(L(1)\) |
\(\approx\) |
\(0.7678822623 - 0.08037093440i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.642 - 0.766i)T \) |
| 11 | \( 1 + (-0.642 + 0.766i)T \) |
| 13 | \( 1 + (-0.342 + 0.939i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (0.342 + 0.939i)T \) |
| 31 | \( 1 + (-0.173 - 0.984i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + (0.939 + 0.342i)T \) |
| 43 | \( 1 + (-0.984 - 0.173i)T \) |
| 47 | \( 1 + (0.173 - 0.984i)T \) |
| 53 | \( 1 + (-0.866 - 0.5i)T \) |
| 59 | \( 1 + (-0.342 + 0.939i)T \) |
| 61 | \( 1 + (-0.984 - 0.173i)T \) |
| 67 | \( 1 + (0.642 + 0.766i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.342 - 0.939i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.173 + 0.984i)T \) |
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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
−19.35537092857468356036763611180, −18.59639820698875992481051732465, −18.17854720018336531556819928649, −17.32891384683493197945010987446, −16.42634713885485521274092382200, −15.81835777862761326304640582624, −15.27985899816941460214047287006, −14.22933770970003264309824297357, −14.13632926468209473183812019389, −12.88260730323162044148542574317, −12.322460190042887288758756104515, −11.56283431525139983369115289159, −10.79051214567309530276109198622, −10.2349311566598809274097592521, −9.578567870009655082732283416360, −8.23116528448129315812992289834, −7.93919901254243213259591932058, −7.32977919516318455928080497316, −6.11056895236765057242113577866, −5.76617740579014070708337155906, −4.70072073473272978855995616794, −3.68892085438331114823756376178, −3.14943036190524285664387660690, −2.3933810831135768765477223338, −1.099876713045097197640564273568,
0.099127686901246320179977654488, 1.30901996674000179711494167935, 2.21672513262107971767319067252, 3.17359010349167560289831457550, 4.22411873422351299569226964149, 4.696485075183646978184981346971, 5.44104471180890313931829254903, 6.467322431946248439096013119795, 7.37115692015561674858854056461, 7.85648151380169217340153511831, 8.698437897017014757696036346187, 9.46505589799225692206270013458, 10.05933780592925232837532843564, 11.05073003656770101671217642523, 11.82008331971796876781871811655, 12.309929620547525782931651834, 13.026951368514345477404688799907, 13.737298277752257948862848543170, 14.70811006861628108327896298866, 15.1996659945877947738295340166, 16.11833397891131072428389372598, 16.51302290013860312753890794931, 17.23169739732496909992469737788, 18.13779771050341902533481703674, 18.69838590383934399162873939333