L(s) = 1 | − i·7-s + (0.309 − 0.951i)11-s + (0.951 − 0.309i)13-s + (0.587 + 0.809i)17-s + (−0.809 + 0.587i)19-s + (−0.951 − 0.309i)23-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (−0.951 + 0.309i)37-s + (−0.309 − 0.951i)41-s − i·43-s + (0.587 − 0.809i)47-s − 49-s + (0.587 − 0.809i)53-s + (−0.309 − 0.951i)59-s + ⋯ |
L(s) = 1 | − i·7-s + (0.309 − 0.951i)11-s + (0.951 − 0.309i)13-s + (0.587 + 0.809i)17-s + (−0.809 + 0.587i)19-s + (−0.951 − 0.309i)23-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (−0.951 + 0.309i)37-s + (−0.309 − 0.951i)41-s − i·43-s + (0.587 − 0.809i)47-s − 49-s + (0.587 − 0.809i)53-s + (−0.309 − 0.951i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7287349768 - 1.232223551i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7287349768 - 1.232223551i\) |
\(L(1)\) |
\(\approx\) |
\(0.9918923711 - 0.3033339400i\) |
\(L(1)\) |
\(\approx\) |
\(0.9918923711 - 0.3033339400i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT \) |
| 11 | \( 1 + (0.309 - 0.951i)T \) |
| 13 | \( 1 + (0.951 - 0.309i)T \) |
| 17 | \( 1 + (0.587 + 0.809i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (-0.951 - 0.309i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.951 + 0.309i)T \) |
| 41 | \( 1 + (-0.309 - 0.951i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.587 - 0.809i)T \) |
| 53 | \( 1 + (0.587 - 0.809i)T \) |
| 59 | \( 1 + (-0.309 - 0.951i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
| 67 | \( 1 + (0.587 + 0.809i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.951 - 0.309i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (0.587 + 0.809i)T \) |
| 89 | \( 1 + (0.309 - 0.951i)T \) |
| 97 | \( 1 + (0.587 - 0.809i)T \) |
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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
−25.529874322748590609203721998902, −24.65660949564244369897319137419, −23.59215771068355355284885392738, −22.76310028379643518941561877378, −21.83874921740595259226296519866, −20.99231980102785227953684658087, −20.09089289650605384572746410463, −18.99376486997275328623408244513, −18.2434070762270591973394848871, −17.38796381544082580073647527500, −16.16127961321185845406996503937, −15.423206853575468975049666955946, −14.49748140753941866904480277882, −13.44608337850967733507871827239, −12.31229608227481125379731427153, −11.68358240374398512123709599821, −10.4545840271834610101345610676, −9.32320377791901151636869602633, −8.598477250215704909922861848138, −7.3123930146541023780913195613, −6.249263758831091230374464214243, −5.20306979334760439097267454879, −4.02166891898858807312595784202, −2.6596875478104846832711175271, −1.49968878264335317125558423657,
0.43255215067138580390200831944, 1.69700042672693662672776545161, 3.47371478448214193458317766406, 4.11165578074757123100971357866, 5.74040073014098977228741826452, 6.51196554912334982984063234768, 7.900070595354452002405857955685, 8.56131966570616164568276272575, 10.040841146439055734780463553841, 10.719375211831826437560516384101, 11.73083526288977193859543009387, 12.97375551491311973036434690033, 13.76033765840412299073321908920, 14.60945625296325636863719938569, 15.81803144255728583458092170496, 16.73001317944069673408874302230, 17.38017237526214586830746626442, 18.67874833011451171937616919809, 19.337245280291960238877596861442, 20.47622781307879110892938126175, 21.1051660772030277322165794800, 22.223130612093949018391664993926, 23.1685029510597095458103962612, 23.864715372902598706430686841766, 24.78044708017380053143247628432