L(s) = 1 | + (−0.546 + 0.837i)3-s + (0.945 + 0.324i)5-s − 7-s + (−0.401 − 0.915i)9-s + (−0.245 + 0.969i)11-s + (0.546 + 0.837i)13-s + (−0.789 + 0.614i)15-s + (−0.0825 + 0.996i)17-s + (0.879 − 0.475i)19-s + (0.546 − 0.837i)21-s + (0.879 − 0.475i)23-s + (0.789 + 0.614i)25-s + (0.986 + 0.164i)27-s + (0.789 − 0.614i)29-s + (0.401 + 0.915i)31-s + ⋯ |
L(s) = 1 | + (−0.546 + 0.837i)3-s + (0.945 + 0.324i)5-s − 7-s + (−0.401 − 0.915i)9-s + (−0.245 + 0.969i)11-s + (0.546 + 0.837i)13-s + (−0.789 + 0.614i)15-s + (−0.0825 + 0.996i)17-s + (0.879 − 0.475i)19-s + (0.546 − 0.837i)21-s + (0.879 − 0.475i)23-s + (0.789 + 0.614i)25-s + (0.986 + 0.164i)27-s + (0.789 − 0.614i)29-s + (0.401 + 0.915i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3800524654 + 1.582182761i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3800524654 + 1.582182761i\) |
\(L(1)\) |
\(\approx\) |
\(0.8589653990 + 0.5026079897i\) |
\(L(1)\) |
\(\approx\) |
\(0.8589653990 + 0.5026079897i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 191 | \( 1 \) |
good | 3 | \( 1 + (-0.546 + 0.837i)T \) |
| 5 | \( 1 + (0.945 + 0.324i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (-0.245 + 0.969i)T \) |
| 13 | \( 1 + (0.546 + 0.837i)T \) |
| 17 | \( 1 + (-0.0825 + 0.996i)T \) |
| 19 | \( 1 + (0.879 - 0.475i)T \) |
| 23 | \( 1 + (0.879 - 0.475i)T \) |
| 29 | \( 1 + (0.789 - 0.614i)T \) |
| 31 | \( 1 + (0.401 + 0.915i)T \) |
| 37 | \( 1 + (-0.401 + 0.915i)T \) |
| 41 | \( 1 + (-0.986 + 0.164i)T \) |
| 43 | \( 1 + (0.677 - 0.735i)T \) |
| 47 | \( 1 + (-0.245 + 0.969i)T \) |
| 53 | \( 1 + (0.245 - 0.969i)T \) |
| 59 | \( 1 + (0.401 + 0.915i)T \) |
| 61 | \( 1 + (-0.0825 - 0.996i)T \) |
| 67 | \( 1 + (0.0825 + 0.996i)T \) |
| 71 | \( 1 + (0.986 - 0.164i)T \) |
| 73 | \( 1 + (0.245 + 0.969i)T \) |
| 79 | \( 1 + (-0.789 - 0.614i)T \) |
| 83 | \( 1 + (0.879 + 0.475i)T \) |
| 89 | \( 1 + (-0.677 - 0.735i)T \) |
| 97 | \( 1 + (-0.401 + 0.915i)T \) |
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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
−22.04130474225533817816549141750, −21.0870110116093640080834917204, −20.19552670416375762606312573006, −19.31448609841698474175885700955, −18.42727989885633058862981546729, −17.99936517443272722949232567772, −16.95659260220913893461139161006, −16.36334214241804068349250490110, −15.62048411752121148807709502576, −14.01213610796039251742957422049, −13.55019997704517788095089722440, −12.93511104631713027644652002743, −12.114434110073756394064077128718, −11.08516617272897750608901739836, −10.26302749608074619676995895041, −9.32247690979990088391055447767, −8.41278588965196586627138661563, −7.33539776116954650914636702927, −6.42480952868761355348000688551, −5.69225537133241269142881325020, −5.133049739761579077643776238335, −3.320864236113034084013685466978, −2.57094169955254699238636375791, −1.16068318556112678827123026054, −0.46625526542785609447591620223,
1.17584888568250229683787668979, 2.557349748175933569414496519756, 3.50214406226045681500219185739, 4.59503461171393540783054623844, 5.46048045390262772075158223941, 6.52726473979840590644173470941, 6.82168458618451037930498892700, 8.63062487647361434767470367945, 9.445087534917486965911563058677, 10.08045067387457748479783505539, 10.67331696573910842911195309256, 11.767085034914763363498006635, 12.67693605910548363176504151681, 13.52373245379674567251441373354, 14.44609705196490867282644031791, 15.39130241649587895336896143246, 15.99048828930972307464811689975, 16.98285803989542666648556457570, 17.48252352161938437021649684539, 18.38247609540815845314862482388, 19.28164763993514982456356927986, 20.36051899563005271368378098908, 21.08815507996314318672147033976, 21.76537875321729013338131800090, 22.505021739302968077885846763800