L(s) = 1 | + (−0.985 + 0.170i)2-s + (−0.809 − 0.587i)3-s + (0.941 − 0.336i)4-s + (0.993 − 0.113i)5-s + (0.897 + 0.441i)6-s + (−0.362 + 0.931i)7-s + (−0.870 + 0.491i)8-s + (0.309 + 0.951i)9-s + (−0.959 + 0.281i)10-s + (−0.959 − 0.281i)12-s + (−0.0285 − 0.999i)13-s + (0.198 − 0.980i)14-s + (−0.870 − 0.491i)15-s + (0.774 − 0.633i)16-s + (0.516 + 0.856i)17-s + (−0.466 − 0.884i)18-s + ⋯ |
L(s) = 1 | + (−0.985 + 0.170i)2-s + (−0.809 − 0.587i)3-s + (0.941 − 0.336i)4-s + (0.993 − 0.113i)5-s + (0.897 + 0.441i)6-s + (−0.362 + 0.931i)7-s + (−0.870 + 0.491i)8-s + (0.309 + 0.951i)9-s + (−0.959 + 0.281i)10-s + (−0.959 − 0.281i)12-s + (−0.0285 − 0.999i)13-s + (0.198 − 0.980i)14-s + (−0.870 − 0.491i)15-s + (0.774 − 0.633i)16-s + (0.516 + 0.856i)17-s + (−0.466 − 0.884i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6196461659 + 0.04512657031i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6196461659 + 0.04512657031i\) |
\(L(1)\) |
\(\approx\) |
\(0.6525091186 + 0.005524492688i\) |
\(L(1)\) |
\(\approx\) |
\(0.6525091186 + 0.005524492688i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (-0.985 + 0.170i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.993 - 0.113i)T \) |
| 7 | \( 1 + (-0.362 + 0.931i)T \) |
| 13 | \( 1 + (-0.0285 - 0.999i)T \) |
| 17 | \( 1 + (0.516 + 0.856i)T \) |
| 19 | \( 1 + (0.0855 + 0.996i)T \) |
| 23 | \( 1 + (0.841 + 0.540i)T \) |
| 29 | \( 1 + (0.974 + 0.226i)T \) |
| 31 | \( 1 + (-0.564 - 0.825i)T \) |
| 37 | \( 1 + (0.610 - 0.791i)T \) |
| 41 | \( 1 + (0.696 - 0.717i)T \) |
| 43 | \( 1 + (0.415 - 0.909i)T \) |
| 47 | \( 1 + (-0.466 + 0.884i)T \) |
| 53 | \( 1 + (0.774 + 0.633i)T \) |
| 59 | \( 1 + (0.696 + 0.717i)T \) |
| 61 | \( 1 + (-0.985 - 0.170i)T \) |
| 67 | \( 1 + (-0.142 + 0.989i)T \) |
| 71 | \( 1 + (-0.921 + 0.389i)T \) |
| 73 | \( 1 + (-0.998 + 0.0570i)T \) |
| 79 | \( 1 + (-0.736 + 0.676i)T \) |
| 83 | \( 1 + (-0.254 - 0.967i)T \) |
| 89 | \( 1 + (-0.654 + 0.755i)T \) |
| 97 | \( 1 + (0.993 + 0.113i)T \) |
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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
−29.08390189755602852105985334967, −28.120553911938897004396169264131, −26.88859532502499259992504212655, −26.360533363620605955820618999799, −25.312351955974780281619413111118, −24.03915355017233480406294678524, −22.85895240026599435754766156335, −21.59433109478307628936243794277, −20.98330647274257776847334669884, −19.821117183808058000769102587451, −18.45823900169702600720398429693, −17.583257424198791156883297133, −16.71995615101163018332653990653, −16.12111372709790259470325324913, −14.54617028376608200168435898573, −13.10349931519920097000048496468, −11.67626841439741620163174713222, −10.70210293427934664554105865241, −9.8328453993645610122484024294, −9.07926477173334210164618087003, −7.06724694297071514100949687460, −6.37486528616251066564365604701, −4.75119443023947297672747074812, −2.98180940970223898431143480769, −1.08007999276501528129520498853,
1.32648672429399046783191691320, 2.6099452721686794090759419198, 5.67427503692102616485324851095, 5.892987033833711545973918257788, 7.37910834143777172330903183486, 8.63776348856496726815711974779, 9.8958961818261578304448010216, 10.762731994531565604477545906045, 12.16771552361853344414360215235, 12.95978361370054696130771461518, 14.65290259438703141524324422386, 15.9447277909552507969650959960, 16.97226725413492340663259925523, 17.751016911921741753396563416031, 18.55117988859630046731124128580, 19.428234875223535389843436668508, 20.92381973674539370914391686658, 21.906559343535864719169966416159, 23.103536210058832759769865181528, 24.41912080910700763233911036880, 25.16018167888141820232983575264, 25.72482726363050921556664792089, 27.38855351446612230892474871683, 28.11452070022652460177921354723, 29.07085123327091425217126062107