L(s) = 1 | + (0.983 + 0.178i)2-s + (0.858 + 0.512i)3-s + (0.936 + 0.351i)4-s + (−0.393 − 0.919i)5-s + (0.753 + 0.657i)6-s + (0.858 + 0.512i)8-s + (0.473 + 0.880i)9-s + (−0.222 − 0.974i)10-s + (0.623 + 0.781i)12-s + (0.983 + 0.178i)13-s + (0.134 − 0.990i)15-s + (0.753 + 0.657i)16-s + (−0.963 − 0.266i)17-s + (0.309 + 0.951i)18-s + (0.309 − 0.951i)19-s + (−0.0448 − 0.998i)20-s + ⋯ |
L(s) = 1 | + (0.983 + 0.178i)2-s + (0.858 + 0.512i)3-s + (0.936 + 0.351i)4-s + (−0.393 − 0.919i)5-s + (0.753 + 0.657i)6-s + (0.858 + 0.512i)8-s + (0.473 + 0.880i)9-s + (−0.222 − 0.974i)10-s + (0.623 + 0.781i)12-s + (0.983 + 0.178i)13-s + (0.134 − 0.990i)15-s + (0.753 + 0.657i)16-s + (−0.963 − 0.266i)17-s + (0.309 + 0.951i)18-s + (0.309 − 0.951i)19-s + (−0.0448 − 0.998i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.270166136 + 0.7700033631i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.270166136 + 0.7700033631i\) |
\(L(1)\) |
\(\approx\) |
\(2.343463323 + 0.4044903959i\) |
\(L(1)\) |
\(\approx\) |
\(2.343463323 + 0.4044903959i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.983 + 0.178i)T \) |
| 3 | \( 1 + (0.858 + 0.512i)T \) |
| 5 | \( 1 + (-0.393 - 0.919i)T \) |
| 13 | \( 1 + (0.983 + 0.178i)T \) |
| 17 | \( 1 + (-0.963 - 0.266i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (0.623 - 0.781i)T \) |
| 29 | \( 1 + (-0.550 + 0.834i)T \) |
| 31 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.550 + 0.834i)T \) |
| 41 | \( 1 + (0.858 + 0.512i)T \) |
| 43 | \( 1 + (-0.222 - 0.974i)T \) |
| 47 | \( 1 + (0.134 + 0.990i)T \) |
| 53 | \( 1 + (-0.963 + 0.266i)T \) |
| 59 | \( 1 + (0.858 - 0.512i)T \) |
| 61 | \( 1 + (-0.963 - 0.266i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.0448 + 0.998i)T \) |
| 73 | \( 1 + (0.134 - 0.990i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (0.983 - 0.178i)T \) |
| 89 | \( 1 + (-0.900 - 0.433i)T \) |
| 97 | \( 1 + (-0.809 - 0.587i)T \) |
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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
−23.205428598715397612125343579182, −22.76471583881769830086501826166, −21.621552882341357235701366257337, −20.910036719430707740373687610264, −20.028671795261906904949401154642, −19.339310647249334901529443553963, −18.61026848603967474570650684628, −17.716470854394477588769314846096, −16.12770596363146559908083612827, −15.39339744665616406409283512456, −14.7347039976643334502706487495, −13.93402181732114273669596546047, −13.25906918205342142536284753765, −12.38543229275463962837891469539, −11.35698795591464146634431110195, −10.67447967944629550391247440019, −9.48772853445037327916354247741, −8.199084540263170800598154570756, −7.333795698968534733118702070823, −6.57456245515934307389999777139, −5.65515981912125700891509054927, −3.9865566222208993227792118169, −3.53061617740631359106803663566, −2.49549259881936386937352812074, −1.5069108381655462163305238120,
1.55492489904953081586491856881, 2.77332517076843315886139810365, 3.78163410581203449841087849308, 4.551983920833514451360162081449, 5.28457515793855786674683132734, 6.6647806336157855698039662567, 7.66602031652713009551257417184, 8.63876243725216206681276899612, 9.26306867032734562818401107764, 10.80192003283885514689348809576, 11.40126034120576566097852590429, 12.73755685895633327653295225519, 13.23977660469012750635544887186, 14.043069215842691499427566365314, 15.05359392242353649521552807736, 15.76405580998811335127713077297, 16.2664546919351353269545877840, 17.2262081400189003527313082642, 18.70113927144539710395772329779, 19.759273240389086568596925996704, 20.42736232605345662117099058780, 20.80250867146999024670182862902, 21.83620339468649996995400986406, 22.50295678772568374474960983394, 23.66906490961002035297100861903