L(s) = 1 | + (0.993 + 0.116i)5-s + (0.893 − 0.448i)7-s + (−0.396 + 0.918i)11-s + (−0.973 + 0.230i)13-s + (0.173 + 0.984i)17-s + (−0.173 + 0.984i)19-s + (0.893 + 0.448i)23-s + (0.973 + 0.230i)25-s + (0.286 + 0.957i)29-s + (−0.835 + 0.549i)31-s + (0.939 − 0.342i)35-s + (0.939 + 0.342i)37-s + (−0.686 − 0.727i)41-s + (−0.597 − 0.802i)43-s + (−0.835 − 0.549i)47-s + ⋯ |
L(s) = 1 | + (0.993 + 0.116i)5-s + (0.893 − 0.448i)7-s + (−0.396 + 0.918i)11-s + (−0.973 + 0.230i)13-s + (0.173 + 0.984i)17-s + (−0.173 + 0.984i)19-s + (0.893 + 0.448i)23-s + (0.973 + 0.230i)25-s + (0.286 + 0.957i)29-s + (−0.835 + 0.549i)31-s + (0.939 − 0.342i)35-s + (0.939 + 0.342i)37-s + (−0.686 − 0.727i)41-s + (−0.597 − 0.802i)43-s + (−0.835 − 0.549i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.605277697 + 0.6558281062i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.605277697 + 0.6558281062i\) |
\(L(1)\) |
\(\approx\) |
\(1.290077178 + 0.1985251542i\) |
\(L(1)\) |
\(\approx\) |
\(1.290077178 + 0.1985251542i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.993 + 0.116i)T \) |
| 7 | \( 1 + (0.893 - 0.448i)T \) |
| 11 | \( 1 + (-0.396 + 0.918i)T \) |
| 13 | \( 1 + (-0.973 + 0.230i)T \) |
| 17 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (-0.173 + 0.984i)T \) |
| 23 | \( 1 + (0.893 + 0.448i)T \) |
| 29 | \( 1 + (0.286 + 0.957i)T \) |
| 31 | \( 1 + (-0.835 + 0.549i)T \) |
| 37 | \( 1 + (0.939 + 0.342i)T \) |
| 41 | \( 1 + (-0.686 - 0.727i)T \) |
| 43 | \( 1 + (-0.597 - 0.802i)T \) |
| 47 | \( 1 + (-0.835 - 0.549i)T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.396 - 0.918i)T \) |
| 61 | \( 1 + (0.0581 - 0.998i)T \) |
| 67 | \( 1 + (0.286 - 0.957i)T \) |
| 71 | \( 1 + (0.766 - 0.642i)T \) |
| 73 | \( 1 + (0.766 + 0.642i)T \) |
| 79 | \( 1 + (-0.686 + 0.727i)T \) |
| 83 | \( 1 + (0.686 - 0.727i)T \) |
| 89 | \( 1 + (0.766 + 0.642i)T \) |
| 97 | \( 1 + (-0.993 + 0.116i)T \) |
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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
−22.60586766600787240473230471724, −21.74297657672727454844000997716, −21.276278119163546910703235194800, −20.5117494196271968677620238972, −19.4723716000531972782759192895, −18.433523185995529412662196669054, −17.919752759940740244054692475377, −17.03510028093208053452197042053, −16.318859164789453351196946964611, −15.08077408324678050082594530814, −14.5013329753770446472038955004, −13.51854119011060295448385379294, −12.91118696257908916041640351873, −11.6695446640074125097979892556, −11.04257402902499451336108057405, −9.942041406062743688098813180894, −9.15964837507480896148513233616, −8.294307309357975390927042185516, −7.28910395557574398233670948432, −6.15764931232600463445555961120, −5.22397463863637537824166147486, −4.694836908825223479049981519962, −2.876937615989675273379946904651, −2.30410101370774734481342345211, −0.89026273390280475116154766645,
1.53081977104589266338021656805, 2.10590975415831144333254464060, 3.51908989484290067834899479227, 4.82697649192047966265716950711, 5.34501044322467055529951208885, 6.64044644925079411615455887674, 7.42792104524099995256332150109, 8.40381539826456854874330576793, 9.51793142683886785461802588701, 10.282631593052222018812403282898, 10.92178697279447013708598501257, 12.2265691481887315791438945197, 12.86850185299096923355350393110, 13.93560150455905657759762987866, 14.631139833034455788496034078565, 15.20097246430277516519086973810, 16.80515358771008036408424461868, 17.09180129299990393128051751097, 17.988295623247074298326777194069, 18.661764377792321352024626969417, 19.89005694806366692243445620082, 20.55436347492467315965537938289, 21.457818887178936655739350212619, 21.86980747114193137601283345512, 23.11264746850300093369238579506