L(s) = 1 | + (0.382 − 0.923i)3-s + (0.923 − 0.382i)5-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.382 − 0.923i)11-s + (0.923 + 0.382i)13-s − i·15-s + i·17-s + (−0.923 − 0.382i)19-s + (0.382 + 0.923i)21-s + (0.707 + 0.707i)23-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s + (−0.382 + 0.923i)29-s − 31-s + ⋯ |
L(s) = 1 | + (0.382 − 0.923i)3-s + (0.923 − 0.382i)5-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.382 − 0.923i)11-s + (0.923 + 0.382i)13-s − i·15-s + i·17-s + (−0.923 − 0.382i)19-s + (0.382 + 0.923i)21-s + (0.707 + 0.707i)23-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s + (−0.382 + 0.923i)29-s − 31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9586907000 - 0.4534269321i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9586907000 - 0.4534269321i\) |
\(L(1)\) |
\(\approx\) |
\(1.103319112 - 0.3385797248i\) |
\(L(1)\) |
\(\approx\) |
\(1.103319112 - 0.3385797248i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (0.382 - 0.923i)T \) |
| 5 | \( 1 + (0.923 - 0.382i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
| 11 | \( 1 + (-0.382 - 0.923i)T \) |
| 13 | \( 1 + (0.923 + 0.382i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (-0.923 - 0.382i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
| 29 | \( 1 + (-0.382 + 0.923i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.923 + 0.382i)T \) |
| 41 | \( 1 + (0.707 + 0.707i)T \) |
| 43 | \( 1 + (0.382 + 0.923i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (-0.382 - 0.923i)T \) |
| 59 | \( 1 + (0.923 - 0.382i)T \) |
| 61 | \( 1 + (0.382 - 0.923i)T \) |
| 67 | \( 1 + (0.382 - 0.923i)T \) |
| 71 | \( 1 + (-0.707 + 0.707i)T \) |
| 73 | \( 1 + (-0.707 - 0.707i)T \) |
| 79 | \( 1 - iT \) |
| 83 | \( 1 + (-0.923 - 0.382i)T \) |
| 89 | \( 1 + (0.707 - 0.707i)T \) |
| 97 | \( 1 - T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−32.73215844500233758450981100895, −31.46357620867303411548691054836, −30.2236644165998725098572944233, −29.08952000091600276096268261521, −28.003298046897955936493752197394, −26.70730903401464501529457082337, −25.796158802108026199878894031294, −25.18654165990104125698088403989, −23.06677202563200077524712833237, −22.44471636748367210645202510909, −20.96518881824040102334049177304, −20.42969700937700515173475737949, −18.878157943525390729449708229648, −17.470298579754257426432358037862, −16.36595331507539501380790301153, −15.166580337377798463414526125530, −13.97358783883417042832449756821, −12.96796119952299669664313030538, −10.78633022486154205693127908843, −10.0724887365345354937158277139, −8.953443959306035956208098964620, −7.11813004944074725412207890767, −5.56690554871849515191828521469, −3.96910493400335712751711719044, −2.498428486412202240768336624240,
1.688359758410706679331512679678, 3.189313832646864378474855163316, 5.6965377329053514766056664882, 6.52943091678980316039062948736, 8.4464489738136899646827164789, 9.21882219366086194265759404554, 11.04589942041792045045910025292, 12.78677290303939034347855293581, 13.237023294488903619044807616737, 14.58301012384432715650466098400, 16.164226387982578871708495214372, 17.469052900381844741388388931726, 18.62530360519801950130414946241, 19.42326223172920093413802511480, 20.94471290841906939370173284783, 21.85502846710737684552901093400, 23.48535085486100316111736550306, 24.39827437036053570347049168176, 25.60836958179665724073823456782, 25.98892277352743353397159398426, 28.05003098386480199185992844359, 29.03937499531797899884259282473, 29.72266435273047508877109248813, 31.0849621364813562537218827219, 32.011096407705070349233394780967