L(s) = 1 | + (0.826 + 0.563i)2-s + (0.623 − 0.781i)3-s + (0.365 + 0.930i)4-s + (−0.988 + 0.149i)5-s + (0.955 − 0.294i)6-s + (0.826 − 0.563i)7-s + (−0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.900 − 0.433i)10-s + (0.623 − 0.781i)11-s + (0.955 + 0.294i)12-s + (−0.988 + 0.149i)13-s + 14-s + (−0.5 + 0.866i)15-s + (−0.733 + 0.680i)16-s + (0.623 + 0.781i)17-s + ⋯ |
L(s) = 1 | + (0.826 + 0.563i)2-s + (0.623 − 0.781i)3-s + (0.365 + 0.930i)4-s + (−0.988 + 0.149i)5-s + (0.955 − 0.294i)6-s + (0.826 − 0.563i)7-s + (−0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.900 − 0.433i)10-s + (0.623 − 0.781i)11-s + (0.955 + 0.294i)12-s + (−0.988 + 0.149i)13-s + 14-s + (−0.5 + 0.866i)15-s + (−0.733 + 0.680i)16-s + (0.623 + 0.781i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.848634210 + 0.03196516127i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.848634210 + 0.03196516127i\) |
\(L(1)\) |
\(\approx\) |
\(1.918390828 + 0.1259884996i\) |
\(L(1)\) |
\(\approx\) |
\(1.918390828 + 0.1259884996i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (0.826 + 0.563i)T \) |
| 3 | \( 1 + (0.623 - 0.781i)T \) |
| 5 | \( 1 + (-0.988 + 0.149i)T \) |
| 7 | \( 1 + (0.826 - 0.563i)T \) |
| 11 | \( 1 + (0.623 - 0.781i)T \) |
| 13 | \( 1 + (-0.988 + 0.149i)T \) |
| 17 | \( 1 + (0.623 + 0.781i)T \) |
| 19 | \( 1 + (0.826 + 0.563i)T \) |
| 23 | \( 1 + (0.623 + 0.781i)T \) |
| 29 | \( 1 + (0.623 - 0.781i)T \) |
| 31 | \( 1 + (-0.988 + 0.149i)T \) |
| 37 | \( 1 + (0.826 - 0.563i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (0.365 - 0.930i)T \) |
| 47 | \( 1 + (0.826 - 0.563i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.0747 - 0.997i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.623 - 0.781i)T \) |
| 71 | \( 1 + (-0.900 - 0.433i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.955 + 0.294i)T \) |
| 83 | \( 1 + (0.365 - 0.930i)T \) |
| 89 | \( 1 + (0.365 + 0.930i)T \) |
| 97 | \( 1 + (-0.222 + 0.974i)T \) |
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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
−21.85518863818265569868965786307, −20.791671841730483843483456965875, −20.32312454494505127812267293009, −19.74254693708621625073514691845, −18.96993225384849182861783618758, −18.03041232193137752006255221412, −16.697067914405329229039293191887, −15.899068322802635917641993803986, −15.12425080536891757598986720660, −14.60990817249708470337073726020, −14.1428711338602694041820201652, −12.76599313675937707990549979360, −12.10820688928783869962647370971, −11.393793709690999560322969619609, −10.67341352314602697450993938544, −9.53142062022568265189687357964, −9.04681464808629188210725827391, −7.74718184842093495735388845432, −7.13650760389858336350892930144, −5.516241264493976429114622568554, −4.62824715778116946890836121565, −4.45054547682471878412414273148, −3.09073184089990267993597063940, −2.57858803877533135125689848557, −1.22098247817729094382039222928,
1.05154337679256717174987205668, 2.34349324209777899315197216761, 3.538897167631391068016561178840, 3.89614904379546328086848188652, 5.12059948907092359948052592303, 6.15167228073085431342085753178, 7.23349563944970742720943620944, 7.63021456132817517590373267419, 8.25254281903578236537014629773, 9.2357904597650223788001731345, 10.82087855565023851611290367780, 11.73008452579732385758276309223, 12.13448732223732271360371640762, 13.09283754929351715918950470857, 14.05797634841343009881202665801, 14.50247344196265764867397576482, 15.05264863328257935848007611672, 16.12692297911136355154664093940, 16.986559385924141261750954593977, 17.63291331333998679056387653329, 18.75705749373016828980282724079, 19.54485609792935342359698813669, 20.12178789789654090042489822290, 21.00808311986925458948447954152, 21.77883885440153341928113460006