L(s) = 1 | + (−0.720 + 0.693i)2-s + (0.190 + 0.981i)3-s + (0.0383 − 0.999i)4-s + (0.264 + 0.964i)5-s + (−0.817 − 0.575i)6-s + (0.988 + 0.152i)7-s + (0.665 + 0.746i)8-s + (−0.927 + 0.373i)9-s + (−0.859 − 0.511i)10-s + (0.896 + 0.443i)11-s + (0.988 − 0.152i)12-s + (0.771 + 0.636i)13-s + (−0.817 + 0.575i)14-s + (−0.896 + 0.443i)15-s + (−0.997 − 0.0765i)16-s + (0.477 + 0.878i)17-s + ⋯ |
L(s) = 1 | + (−0.720 + 0.693i)2-s + (0.190 + 0.981i)3-s + (0.0383 − 0.999i)4-s + (0.264 + 0.964i)5-s + (−0.817 − 0.575i)6-s + (0.988 + 0.152i)7-s + (0.665 + 0.746i)8-s + (−0.927 + 0.373i)9-s + (−0.859 − 0.511i)10-s + (0.896 + 0.443i)11-s + (0.988 − 0.152i)12-s + (0.771 + 0.636i)13-s + (−0.817 + 0.575i)14-s + (−0.896 + 0.443i)15-s + (−0.997 − 0.0765i)16-s + (0.477 + 0.878i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2859344506 + 1.376320621i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2859344506 + 1.376320621i\) |
\(L(1)\) |
\(\approx\) |
\(0.6557267517 + 0.7355202488i\) |
\(L(1)\) |
\(\approx\) |
\(0.6557267517 + 0.7355202488i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 \) |
good | 2 | \( 1 + (-0.720 + 0.693i)T \) |
| 3 | \( 1 + (0.190 + 0.981i)T \) |
| 5 | \( 1 + (0.264 + 0.964i)T \) |
| 7 | \( 1 + (0.988 + 0.152i)T \) |
| 11 | \( 1 + (0.896 + 0.443i)T \) |
| 13 | \( 1 + (0.771 + 0.636i)T \) |
| 17 | \( 1 + (0.477 + 0.878i)T \) |
| 19 | \( 1 + (0.114 - 0.993i)T \) |
| 23 | \( 1 + (-0.409 - 0.912i)T \) |
| 29 | \( 1 + (-0.973 - 0.227i)T \) |
| 31 | \( 1 + (-0.665 + 0.746i)T \) |
| 37 | \( 1 + (-0.927 - 0.373i)T \) |
| 41 | \( 1 + (0.720 + 0.693i)T \) |
| 43 | \( 1 + (0.543 - 0.839i)T \) |
| 47 | \( 1 + (-0.338 - 0.941i)T \) |
| 53 | \( 1 + (-0.338 + 0.941i)T \) |
| 59 | \( 1 + (0.606 - 0.795i)T \) |
| 61 | \( 1 + (0.953 - 0.301i)T \) |
| 67 | \( 1 + (0.997 + 0.0765i)T \) |
| 71 | \( 1 + (-0.988 + 0.152i)T \) |
| 73 | \( 1 + (0.859 + 0.511i)T \) |
| 79 | \( 1 + (-0.0383 + 0.999i)T \) |
| 89 | \( 1 + (-0.817 - 0.575i)T \) |
| 97 | \( 1 + (-0.817 + 0.575i)T \) |
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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
−29.816502653071470564536760138509, −29.37993112475318600322960547882, −27.89791038738443284906132914414, −27.44219451345237254524444453302, −25.74011194530799927793114418415, −24.918061484501364431915262935342, −24.069146342605756204614341632264, −22.56927779452396031881430499650, −20.90028203465136466344998496031, −20.45467997455804031374334171844, −19.27565732569411954023999352886, −18.13760115149078027116112119327, −17.36832741810317980412294413306, −16.33627381525352000276474216451, −14.20850276595069394501164987569, −13.19494695831424417284988483679, −12.0621126531550075287737407945, −11.23206034673879110579627315224, −9.42723574291964363364215275801, −8.39164755363132457154942925886, −7.56437116984798145358578354480, −5.67432627902738515414659905240, −3.68339661287817208966461569886, −1.76159914906164494965284549698, −0.91411569517785315248774745329,
1.953779300765316683970829727888, 4.06576333104402680917694378844, 5.577965554891443489808384628152, 6.88506810264579936089376270189, 8.38629502625052400767558106065, 9.39478133638089856666968393128, 10.626011887760212335827231834, 11.36981897488835710665086496738, 14.12279343838083839134728606044, 14.62339045966411681296951387084, 15.58190148102541277872178079820, 16.91404887100648969930698453290, 17.78787777438188987067350832997, 18.949044696913181936548743415703, 20.17874090118287567641210402613, 21.42385553148890427944527637080, 22.48133674289628287678331642103, 23.68832727746851223497265517960, 25.03257129641972164176827232714, 26.04157321816436897650573947653, 26.633954532586687431512885365828, 27.812911358699719686113339849455, 28.3299660287791781506143433458, 30.11366761857663233951419550199, 31.09932156476368818786931233065