L(s) = 1 | + (0.978 + 0.207i)2-s + (0.5 + 0.866i)3-s + (0.913 + 0.406i)4-s + (0.309 + 0.951i)6-s + (0.809 + 0.587i)8-s + (−0.5 + 0.866i)9-s + (0.104 + 0.994i)12-s − 13-s + (0.669 + 0.743i)16-s + (−0.669 + 0.743i)17-s + (−0.669 + 0.743i)18-s + (0.913 − 0.406i)19-s + (−0.913 + 0.406i)23-s + (−0.104 + 0.994i)24-s + (−0.978 − 0.207i)26-s − 27-s + ⋯ |
L(s) = 1 | + (0.978 + 0.207i)2-s + (0.5 + 0.866i)3-s + (0.913 + 0.406i)4-s + (0.309 + 0.951i)6-s + (0.809 + 0.587i)8-s + (−0.5 + 0.866i)9-s + (0.104 + 0.994i)12-s − 13-s + (0.669 + 0.743i)16-s + (−0.669 + 0.743i)17-s + (−0.669 + 0.743i)18-s + (0.913 − 0.406i)19-s + (−0.913 + 0.406i)23-s + (−0.104 + 0.994i)24-s + (−0.978 − 0.207i)26-s − 27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9093916337 + 2.918886164i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9093916337 + 2.918886164i\) |
\(L(1)\) |
\(\approx\) |
\(1.640005480 + 1.174362850i\) |
\(L(1)\) |
\(\approx\) |
\(1.640005480 + 1.174362850i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (-0.669 + 0.743i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (-0.913 + 0.406i)T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.913 + 0.406i)T \) |
| 37 | \( 1 + (-0.913 + 0.406i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.978 - 0.207i)T \) |
| 59 | \( 1 + (-0.104 + 0.994i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.978 - 0.207i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.978 - 0.207i)T \) |
| 79 | \( 1 + (0.669 + 0.743i)T \) |
| 83 | \( 1 + (-0.309 - 0.951i)T \) |
| 89 | \( 1 + (0.913 - 0.406i)T \) |
| 97 | \( 1 + (-0.309 + 0.951i)T \) |
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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
−19.92087460895153112823681220232, −19.15023931238909741636685245699, −18.5219916399930814215823637323, −17.61000418323949176419841011232, −16.80434272270750815781038542119, −15.77765266800529254793428814127, −15.210683091249494833784086042264, −14.218385240541134511316674331705, −13.95667634868471200800263425114, −13.16800125096591394629775653843, −12.31209756948243737256468905139, −11.93178377391651429864564110502, −11.13250722972680566513802233687, −9.9984462514779066477431594972, −9.368641226969922248739797359958, −8.16435865115212345424656780594, −7.42557571170887174980788291484, −6.84039100707418299499571997161, −5.9654158426430839430009081960, −5.17515510865485819441583048448, −4.19087711417985567961582489542, −3.31669418557774963236353239705, −2.42364842802628664659033833329, −1.909329833251040901247360709721, −0.615317674816055835977507427342,
1.74268864054243259003774711413, 2.58824701113956681960503817489, 3.390728337411024997230009154012, 4.13734151497101554829039626782, 4.944489015708481615759880079124, 5.50148018749939401777578017985, 6.57361053452853827045892727479, 7.444899366357443649398883376474, 8.1840823298745914742494260857, 9.069431723469779653938627385161, 10.04091216075323676162720539473, 10.61462029724071047770291118927, 11.60461382502916409088345515884, 12.13587257548621847477187128326, 13.285643535576017944312138634835, 13.71162766181338438966501275805, 14.5892329569459483252622082288, 15.12413421933275234806142903042, 15.70868890000069179537743583436, 16.488681918051732572076629351440, 17.099013809594418442699802789023, 17.9751421753512089487114647847, 19.29309393941419509561510363503, 19.977358340413153138644931682045, 20.24049314958831105558624492386