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
−22.505021739302968077885846763800, −21.76537875321729013338131800090, −21.08815507996314318672147033976, −20.36051899563005271368378098908, −19.28164763993514982456356927986, −18.38247609540815845314862482388, −17.48252352161938437021649684539, −16.98285803989542666648556457570, −15.99048828930972307464811689975, −15.39130241649587895336896143246, −14.44609705196490867282644031791, −13.52373245379674567251441373354, −12.67693605910548363176504151681, −11.767085034914763363498006635, −10.67331696573910842911195309256, −10.08045067387457748479783505539, −9.445087534917486965911563058677, −8.63062487647361434767470367945, −6.82168458618451037930498892700, −6.52726473979840590644173470941, −5.46048045390262772075158223941, −4.59503461171393540783054623844, −3.50214406226045681500219185739, −2.557349748175933569414496519756, −1.17584888568250229683787668979,
0.46625526542785609447591620223, 1.16068318556112678827123026054, 2.57094169955254699238636375791, 3.320864236113034084013685466978, 5.133049739761579077643776238335, 5.69225537133241269142881325020, 6.42480952868761355348000688551, 7.33539776116954650914636702927, 8.41278588965196586627138661563, 9.32247690979990088391055447767, 10.26302749608074619676995895041, 11.08516617272897750608901739836, 12.114434110073756394064077128718, 12.93511104631713027644652002743, 13.55019997704517788095089722440, 14.01213610796039251742957422049, 15.62048411752121148807709502576, 16.36334214241804068349250490110, 16.95659260220913893461139161006, 17.99936517443272722949232567772, 18.42727989885633058862981546729, 19.31448609841698474175885700955, 20.19552670416375762606312573006, 21.0870110116093640080834917204, 22.04130474225533817816549141750