L(s) = 1 | + (0.142 − 0.989i)3-s + (−0.755 + 0.654i)7-s + (−0.959 − 0.281i)9-s + (0.540 − 0.841i)11-s + (−0.654 + 0.755i)13-s + (0.909 − 0.415i)17-s + (−0.909 − 0.415i)19-s + (0.540 + 0.841i)21-s + (−0.415 + 0.909i)27-s + (−0.909 + 0.415i)29-s + (0.142 + 0.989i)31-s + (−0.755 − 0.654i)33-s + (−0.959 − 0.281i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯ |
L(s) = 1 | + (0.142 − 0.989i)3-s + (−0.755 + 0.654i)7-s + (−0.959 − 0.281i)9-s + (0.540 − 0.841i)11-s + (−0.654 + 0.755i)13-s + (0.909 − 0.415i)17-s + (−0.909 − 0.415i)19-s + (0.540 + 0.841i)21-s + (−0.415 + 0.909i)27-s + (−0.909 + 0.415i)29-s + (0.142 + 0.989i)31-s + (−0.755 − 0.654i)33-s + (−0.959 − 0.281i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6670276646 + 0.3774331471i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6670276646 + 0.3774331471i\) |
\(L(1)\) |
\(\approx\) |
\(0.8478499374 - 0.1465151841i\) |
\(L(1)\) |
\(\approx\) |
\(0.8478499374 - 0.1465151841i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.755 + 0.654i)T \) |
| 11 | \( 1 + (0.540 - 0.841i)T \) |
| 13 | \( 1 + (-0.654 + 0.755i)T \) |
| 17 | \( 1 + (0.909 - 0.415i)T \) |
| 19 | \( 1 + (-0.909 - 0.415i)T \) |
| 29 | \( 1 + (-0.909 + 0.415i)T \) |
| 31 | \( 1 + (0.142 + 0.989i)T \) |
| 37 | \( 1 + (-0.959 - 0.281i)T \) |
| 41 | \( 1 + (0.959 - 0.281i)T \) |
| 43 | \( 1 + (-0.142 + 0.989i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.654 + 0.755i)T \) |
| 59 | \( 1 + (-0.755 - 0.654i)T \) |
| 61 | \( 1 + (-0.989 + 0.142i)T \) |
| 67 | \( 1 + (0.841 - 0.540i)T \) |
| 71 | \( 1 + (0.841 - 0.540i)T \) |
| 73 | \( 1 + (-0.909 - 0.415i)T \) |
| 79 | \( 1 + (-0.654 + 0.755i)T \) |
| 83 | \( 1 + (0.959 + 0.281i)T \) |
| 89 | \( 1 + (-0.142 + 0.989i)T \) |
| 97 | \( 1 + (0.281 + 0.959i)T \) |
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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
−20.13991528599895305708264743410, −19.45643421899040184329808683100, −18.767329568671818583972938438152, −17.41768754286815756193650045423, −17.03314155091312618210258534923, −16.49346741811710853577244972685, −15.46124987143560783192805359233, −14.95137372813765181959020653239, −14.338384670995488316994921514178, −13.36490146304569827149433747191, −12.578542811830569917312656944982, −11.84767061346686521766096087247, −10.77410564390430564611405612934, −10.07063302169979293881748631815, −9.78572414097445438069927680816, −8.8346873344245066288338450654, −7.882617805779427954548761940661, −7.1351521675532692015894371871, −6.08299358069214185920084355085, −5.35913524785295163354875409394, −4.27437647767983463988815544246, −3.81122243694209850088705243621, −2.91813425408170396356464127357, −1.865624980640104334116180818976, −0.28360770535151335588409107163,
1.0545814829176332700728725009, 2.0977405676707534669823397182, 2.91461839343750523726433102133, 3.65769910706991443928328481238, 4.990281107642928610720524134873, 5.91475650062759125894884312744, 6.50667582227110008287212713290, 7.22650515835757850181558518965, 8.11298620814298454662217905787, 9.103258544331057399889256388334, 9.30941940386703414430464139885, 10.67338169818659787170996528420, 11.49408337933313874528116502089, 12.30177745224451234773353477158, 12.62802008393952043905725999908, 13.65255417815611641391255903692, 14.24698938253337326554308732703, 14.905130260271930195232637933237, 15.98657498490833881858189332732, 16.688561854767163383473671404187, 17.2980640300813854632151604205, 18.29271210536721858077473301823, 18.92486242282403120578298410023, 19.36890885954851898388805728333, 19.90768886806256994222024588001