L(s) = 1 | + (0.978 − 0.207i)2-s + (0.913 − 0.406i)4-s + (−0.978 − 0.207i)5-s + (0.104 − 0.994i)7-s + (0.809 − 0.587i)8-s − 10-s + (−0.669 − 0.743i)13-s + (−0.104 − 0.994i)14-s + (0.669 − 0.743i)16-s + (−0.309 − 0.951i)17-s + (0.809 − 0.587i)19-s + (−0.978 + 0.207i)20-s + (−0.5 + 0.866i)23-s + (0.913 + 0.406i)25-s + (−0.809 − 0.587i)26-s + ⋯ |
L(s) = 1 | + (0.978 − 0.207i)2-s + (0.913 − 0.406i)4-s + (−0.978 − 0.207i)5-s + (0.104 − 0.994i)7-s + (0.809 − 0.587i)8-s − 10-s + (−0.669 − 0.743i)13-s + (−0.104 − 0.994i)14-s + (0.669 − 0.743i)16-s + (−0.309 − 0.951i)17-s + (0.809 − 0.587i)19-s + (−0.978 + 0.207i)20-s + (−0.5 + 0.866i)23-s + (0.913 + 0.406i)25-s + (−0.809 − 0.587i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.530419971 - 1.890451642i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.530419971 - 1.890451642i\) |
\(L(1)\) |
\(\approx\) |
\(1.485251565 - 0.7064357895i\) |
\(L(1)\) |
\(\approx\) |
\(1.485251565 - 0.7064357895i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.978 - 0.207i)T \) |
| 5 | \( 1 + (-0.978 - 0.207i)T \) |
| 7 | \( 1 + (0.104 - 0.994i)T \) |
| 13 | \( 1 + (-0.669 - 0.743i)T \) |
| 17 | \( 1 + (-0.309 - 0.951i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.104 - 0.994i)T \) |
| 31 | \( 1 + (0.669 + 0.743i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.104 + 0.994i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.913 + 0.406i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.913 - 0.406i)T \) |
| 61 | \( 1 + (-0.669 + 0.743i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.809 + 0.587i)T \) |
| 79 | \( 1 + (0.978 - 0.207i)T \) |
| 83 | \( 1 + (-0.669 + 0.743i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (-0.978 + 0.207i)T \) |
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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
−30.43815215317487353427194141753, −29.0989928397575955167718298411, −28.13065356572815161824718219779, −26.777900780645909184311699706435, −25.76330189118238865306186344535, −24.4138213773360898023393682733, −23.951671798897805031452622389401, −22.56969368550946003975984738243, −21.97759371039192211279509592556, −20.744524837957469785687504477695, −19.59882111371905839465330845559, −18.583937965114269107201041899987, −16.89518844307747147261253961477, −15.7825596303434852697258862496, −15.009147077473239752295083979890, −14.03756948380809655468316086950, −12.33424964893438318394494510098, −11.96808111124175300413407835009, −10.628190582912525289899576667187, −8.66808625269216187938655447057, −7.48509828114418043375574791671, −6.2393897201326261443429484243, −4.86714724406179267190520228852, −3.653708916653056946406869867146, −2.2297798365410413439434505296,
0.79717358796301082575908138126, 2.94173538294176928693213379062, 4.16873926272240758094949901037, 5.18933927198405343557375970218, 7.00392984426408203719851264590, 7.79706118782988757832823075112, 9.86214145524213027298455066374, 11.13966135305088334847810043354, 11.985370943916836812652860972355, 13.21299043950997158106687683780, 14.19769653086180469162286691522, 15.46075364787998112125034085905, 16.21415743331869900342637465884, 17.62754194283285558334244982609, 19.45543991978897385869684589520, 20.00143316743260681821919638088, 20.94640816576623848632129469644, 22.42265962717647964788769525311, 23.07913076704178767097704759874, 24.06931280233294703973005945, 24.86322797461540541173924832239, 26.44753117175184966068590256688, 27.409434314666415989046273149142, 28.61052982814677727298064294483, 29.76464285580526419312841134952