L(s) = 1 | + (−0.458 − 0.888i)2-s + (0.362 − 0.932i)3-s + (−0.579 + 0.815i)4-s + (0.977 + 0.210i)5-s + (−0.994 + 0.105i)6-s + (−0.192 + 0.981i)7-s + (0.990 + 0.140i)8-s + (−0.737 − 0.675i)9-s + (−0.261 − 0.965i)10-s + (0.760 + 0.648i)11-s + (0.550 + 0.835i)12-s + (0.607 − 0.794i)13-s + (0.960 − 0.278i)14-s + (0.550 − 0.835i)15-s + (−0.329 − 0.944i)16-s + (0.844 − 0.535i)17-s + ⋯ |
L(s) = 1 | + (−0.458 − 0.888i)2-s + (0.362 − 0.932i)3-s + (−0.579 + 0.815i)4-s + (0.977 + 0.210i)5-s + (−0.994 + 0.105i)6-s + (−0.192 + 0.981i)7-s + (0.990 + 0.140i)8-s + (−0.737 − 0.675i)9-s + (−0.261 − 0.965i)10-s + (0.760 + 0.648i)11-s + (0.550 + 0.835i)12-s + (0.607 − 0.794i)13-s + (0.960 − 0.278i)14-s + (0.550 − 0.835i)15-s + (−0.329 − 0.944i)16-s + (0.844 − 0.535i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8932553437 - 0.7316539966i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8932553437 - 0.7316539966i\) |
\(L(1)\) |
\(\approx\) |
\(0.9186589903 - 0.5329794957i\) |
\(L(1)\) |
\(\approx\) |
\(0.9186589903 - 0.5329794957i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 179 | \( 1 \) |
good | 2 | \( 1 + (-0.458 - 0.888i)T \) |
| 3 | \( 1 + (0.362 - 0.932i)T \) |
| 5 | \( 1 + (0.977 + 0.210i)T \) |
| 7 | \( 1 + (-0.192 + 0.981i)T \) |
| 11 | \( 1 + (0.760 + 0.648i)T \) |
| 13 | \( 1 + (0.607 - 0.794i)T \) |
| 17 | \( 1 + (0.844 - 0.535i)T \) |
| 19 | \( 1 + (-0.825 + 0.564i)T \) |
| 23 | \( 1 + (0.997 + 0.0705i)T \) |
| 29 | \( 1 + (-0.949 - 0.312i)T \) |
| 31 | \( 1 + (0.295 - 0.955i)T \) |
| 37 | \( 1 + (-0.949 + 0.312i)T \) |
| 41 | \( 1 + (-0.999 + 0.0352i)T \) |
| 43 | \( 1 + (0.911 - 0.411i)T \) |
| 47 | \( 1 + (-0.123 + 0.992i)T \) |
| 53 | \( 1 + (-0.896 - 0.442i)T \) |
| 59 | \( 1 + (0.662 - 0.749i)T \) |
| 61 | \( 1 + (-0.969 - 0.244i)T \) |
| 67 | \( 1 + (0.990 - 0.140i)T \) |
| 71 | \( 1 + (0.158 - 0.987i)T \) |
| 73 | \( 1 + (0.0176 + 0.999i)T \) |
| 79 | \( 1 + (-0.863 + 0.505i)T \) |
| 83 | \( 1 + (0.427 + 0.904i)T \) |
| 89 | \( 1 + (-0.458 + 0.888i)T \) |
| 97 | \( 1 + (0.489 + 0.871i)T \) |
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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
−27.36771461859364357564880587565, −26.35619755314328053792536796925, −25.85829068355012759912744869837, −24.993015484676394896384441443129, −23.862009449412836506035887298682, −22.85767501661414751023059905111, −21.74082632330878504863191534504, −20.91624104245763369789932308976, −19.70609006193400860840141269881, −18.85036456205216980621019522536, −17.201535858364722727749523820160, −16.907814283811777369361598752094, −16.046909686833499987479682733842, −14.67515097226641259522543416995, −14.04299647786264146006859130828, −13.2285115345677555165250869289, −10.96377523603920895920875098796, −10.23668864652267168447897425854, −9.158571089115143238475809010635, −8.59163788189299933817281262949, −6.96587857207549534208541884410, −5.9464983399336658805696225485, −4.75155106840732098432661728846, −3.59999927378035603607505782064, −1.44492201007543501665982057098,
1.39985511796797128430300921550, 2.35724355856289104907009541040, 3.3971731836778185737728419181, 5.451189940280048494174742861, 6.64063235715585939991879691623, 8.00028650568613877813101966158, 9.058354627875397873603492965691, 9.78448119879839431616124503732, 11.19603931391992520967943783522, 12.35591687362380896218801848919, 12.93260552756832876033819455505, 14.001269488118734353618308984912, 15.05433442889324616979023242802, 16.97093414577245650779365358623, 17.64758966953529100126663460640, 18.69086188520873301724575946522, 19.018492510672953712081338182738, 20.4884576023983751840461789748, 21.001839689044740431945787561937, 22.358246016299886866834280822954, 22.92879168389949875890158287173, 24.69744688676141435364737648601, 25.534549646838397294962509291275, 25.74551968993436834195643805493, 27.41891456634927379614452321431