L(s) = 1 | + (−0.755 + 0.654i)3-s + (0.909 + 0.415i)7-s + (0.142 − 0.989i)9-s + (0.959 + 0.281i)11-s + (−0.909 + 0.415i)13-s + (0.540 − 0.841i)17-s + (0.841 − 0.540i)19-s + (−0.959 + 0.281i)21-s + (0.540 + 0.841i)27-s + (−0.841 − 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.909 + 0.415i)33-s + (0.989 + 0.142i)37-s + (0.415 − 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯ |
L(s) = 1 | + (−0.755 + 0.654i)3-s + (0.909 + 0.415i)7-s + (0.142 − 0.989i)9-s + (0.959 + 0.281i)11-s + (−0.909 + 0.415i)13-s + (0.540 − 0.841i)17-s + (0.841 − 0.540i)19-s + (−0.959 + 0.281i)21-s + (0.540 + 0.841i)27-s + (−0.841 − 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.909 + 0.415i)33-s + (0.989 + 0.142i)37-s + (0.415 − 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.131323228 + 0.4096645956i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.131323228 + 0.4096645956i\) |
\(L(1)\) |
\(\approx\) |
\(0.9677973522 + 0.2156906400i\) |
\(L(1)\) |
\(\approx\) |
\(0.9677973522 + 0.2156906400i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.755 + 0.654i)T \) |
| 7 | \( 1 + (0.909 + 0.415i)T \) |
| 11 | \( 1 + (0.959 + 0.281i)T \) |
| 13 | \( 1 + (-0.909 + 0.415i)T \) |
| 17 | \( 1 + (0.540 - 0.841i)T \) |
| 19 | \( 1 + (0.841 - 0.540i)T \) |
| 29 | \( 1 + (-0.841 - 0.540i)T \) |
| 31 | \( 1 + (0.654 - 0.755i)T \) |
| 37 | \( 1 + (0.989 + 0.142i)T \) |
| 41 | \( 1 + (-0.142 - 0.989i)T \) |
| 43 | \( 1 + (-0.755 + 0.654i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.909 + 0.415i)T \) |
| 59 | \( 1 + (0.415 + 0.909i)T \) |
| 61 | \( 1 + (-0.654 + 0.755i)T \) |
| 67 | \( 1 + (-0.281 - 0.959i)T \) |
| 71 | \( 1 + (0.959 - 0.281i)T \) |
| 73 | \( 1 + (0.540 + 0.841i)T \) |
| 79 | \( 1 + (0.415 + 0.909i)T \) |
| 83 | \( 1 + (0.989 + 0.142i)T \) |
| 89 | \( 1 + (0.654 + 0.755i)T \) |
| 97 | \( 1 + (-0.989 + 0.142i)T \) |
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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
−23.80954673441708449986854565765, −23.113466361018630302059796142112, −22.1320946113666729068776488967, −21.534555879189355871063185462405, −20.24446611196264950785458250190, −19.55218600096365451228911098405, −18.54531176334088987342775930356, −17.75564939453876562935283503919, −16.97580595421901522658382573951, −16.47776147353159346759954047858, −14.94680044840307409607562016143, −14.26920653283905111076281282810, −13.30147038714153158975678793867, −12.22432003138162593690395606876, −11.66560798228457038450021839534, −10.71850237248778886599106896047, −9.8237998497547826783168900275, −8.35627378709693603503942800548, −7.609071931289041503561015779762, −6.70921428610022774482956540224, −5.614668627304480863120136876500, −4.81584136535014536530660263545, −3.55024498324236310540730870207, −1.91404266692714393037663743582, −1.01167737338526899259834013484,
1.09026599900881623801965333207, 2.56387712402841649950710618548, 4.03837170975589990148610026684, 4.848744529912726992616328422879, 5.64934848640094328879920654643, 6.81967030228300658926628653659, 7.799059822748071918829255636062, 9.29367472884245864821713665067, 9.617968008321789537609761461541, 10.96552252532648910082227646065, 11.80173161015665886114391053718, 12.11056907188793709704359382326, 13.71106302527207414798447976965, 14.71371864417809883541198318506, 15.24540300959823471960239797640, 16.419039775638133426232730276161, 17.092217675090483461348935487309, 17.85231924597724718896032126529, 18.69270683967238046240760310228, 19.91929083530694044905703421096, 20.774708572420228673418624919726, 21.55343473543387671079470887371, 22.32722005651540803736885426199, 22.91863702692075357769100929666, 24.2052283353943146231828431920