L(s) = 1 | + (−0.508 + 0.861i)2-s + (−0.743 + 0.669i)3-s + (−0.483 − 0.875i)4-s + (−0.198 − 0.980i)6-s + (0.999 + 0.0285i)8-s + (0.104 − 0.994i)9-s + (0.945 + 0.327i)12-s + (0.996 − 0.0855i)13-s + (−0.532 + 0.846i)16-s + (0.893 + 0.449i)17-s + (0.803 + 0.595i)18-s + (−0.710 + 0.703i)19-s + (0.371 + 0.928i)23-s + (−0.761 + 0.647i)24-s + (−0.432 + 0.901i)26-s + (0.587 + 0.809i)27-s + ⋯ |
L(s) = 1 | + (−0.508 + 0.861i)2-s + (−0.743 + 0.669i)3-s + (−0.483 − 0.875i)4-s + (−0.198 − 0.980i)6-s + (0.999 + 0.0285i)8-s + (0.104 − 0.994i)9-s + (0.945 + 0.327i)12-s + (0.996 − 0.0855i)13-s + (−0.532 + 0.846i)16-s + (0.893 + 0.449i)17-s + (0.803 + 0.595i)18-s + (−0.710 + 0.703i)19-s + (0.371 + 0.928i)23-s + (−0.761 + 0.647i)24-s + (−0.432 + 0.901i)26-s + (0.587 + 0.809i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2716652074 + 0.9824960970i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2716652074 + 0.9824960970i\) |
\(L(1)\) |
\(\approx\) |
\(0.5587392634 + 0.4448652422i\) |
\(L(1)\) |
\(\approx\) |
\(0.5587392634 + 0.4448652422i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.508 + 0.861i)T \) |
| 3 | \( 1 + (-0.743 + 0.669i)T \) |
| 13 | \( 1 + (0.996 - 0.0855i)T \) |
| 17 | \( 1 + (0.893 + 0.449i)T \) |
| 19 | \( 1 + (-0.710 + 0.703i)T \) |
| 23 | \( 1 + (0.371 + 0.928i)T \) |
| 29 | \( 1 + (0.774 + 0.633i)T \) |
| 31 | \( 1 + (-0.290 + 0.956i)T \) |
| 37 | \( 1 + (0.992 - 0.123i)T \) |
| 41 | \( 1 + (0.736 + 0.676i)T \) |
| 43 | \( 1 + (0.281 + 0.959i)T \) |
| 47 | \( 1 + (-0.803 + 0.595i)T \) |
| 53 | \( 1 + (0.846 - 0.532i)T \) |
| 59 | \( 1 + (0.217 + 0.976i)T \) |
| 61 | \( 1 + (-0.00951 - 0.999i)T \) |
| 67 | \( 1 + (0.814 - 0.580i)T \) |
| 71 | \( 1 + (-0.362 + 0.931i)T \) |
| 73 | \( 1 + (-0.938 - 0.345i)T \) |
| 79 | \( 1 + (0.380 - 0.924i)T \) |
| 83 | \( 1 + (0.717 - 0.696i)T \) |
| 89 | \( 1 + (-0.0475 + 0.998i)T \) |
| 97 | \( 1 + (0.336 - 0.941i)T \) |
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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
−18.21269676595828302279537311758, −17.58200070437779585887562219863, −16.87260107959433139513568621737, −16.42459319395415999021439681767, −15.63951240547328206255291062545, −14.510057647037584543740313922398, −13.64861900370851233466703778186, −13.17003801793677507735048632454, −12.5330191704541758770077097599, −11.82769407089997458716305996704, −11.288442388403718641199811870700, −10.66105629178903636626068667475, −10.0668063387363772480019546371, −9.11577246011242702357873006737, −8.41722220140142054799918671514, −7.76846847797932807362208696993, −6.99411471343697428565960732624, −6.25883120193767112583867686577, −5.4017004940545148387866870946, −4.518730502196319446072723394705, −3.834518526376877554492370598066, −2.68551290673693515981142507950, −2.167618882529539859148139702561, −1.06609006260366986265450770962, −0.54552017454447940215333938654,
0.961982285304516776742581557924, 1.535174912031659005159694377896, 3.16282774751896595708173265974, 3.93954774588702025188055225058, 4.707015815847949400882350401921, 5.47793552263561507252972854016, 6.08316026425217014439853446958, 6.56866046340178744286569297341, 7.56299199953699071840409649020, 8.28448977850804242463511153057, 8.9911884994572781176797698862, 9.70218236918559173511784370309, 10.3602504548714137111389163264, 10.89920707500174872585229202198, 11.59532931886231612038629820837, 12.61570923814339125494840246662, 13.200073736666487301320427194983, 14.34169377580129369623480044, 14.669689011646645817112748928919, 15.46539681160190682207089279175, 16.20038302339944772863207022577, 16.435911676562376765963363978838, 17.222256360032253821799063018919, 17.92264564204699571519717242226, 18.28134606167499869103939876743