L(s) = 1 | + (0.978 − 0.207i)2-s + (0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (−0.104 − 0.994i)6-s − 7-s + (0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (0.309 − 0.951i)11-s + (−0.309 − 0.951i)12-s + (0.978 + 0.207i)13-s + (−0.978 + 0.207i)14-s + (0.669 − 0.743i)16-s + (−0.913 − 0.406i)17-s − 18-s + (−0.104 + 0.994i)21-s + (0.104 − 0.994i)22-s + ⋯ |
L(s) = 1 | + (0.978 − 0.207i)2-s + (0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (−0.104 − 0.994i)6-s − 7-s + (0.809 − 0.587i)8-s + (−0.978 − 0.207i)9-s + (0.309 − 0.951i)11-s + (−0.309 − 0.951i)12-s + (0.978 + 0.207i)13-s + (−0.978 + 0.207i)14-s + (0.669 − 0.743i)16-s + (−0.913 − 0.406i)17-s − 18-s + (−0.104 + 0.994i)21-s + (0.104 − 0.994i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.028657276 - 1.959486534i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.028657276 - 1.959486534i\) |
\(L(1)\) |
\(\approx\) |
\(1.416488218 - 1.006690479i\) |
\(L(1)\) |
\(\approx\) |
\(1.416488218 - 1.006690479i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.978 - 0.207i)T \) |
| 3 | \( 1 + (0.104 - 0.994i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (0.309 - 0.951i)T \) |
| 13 | \( 1 + (0.978 + 0.207i)T \) |
| 17 | \( 1 + (-0.913 - 0.406i)T \) |
| 23 | \( 1 + (-0.669 - 0.743i)T \) |
| 29 | \( 1 + (0.913 - 0.406i)T \) |
| 31 | \( 1 + (-0.809 + 0.587i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (0.669 - 0.743i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.913 + 0.406i)T \) |
| 53 | \( 1 + (-0.913 + 0.406i)T \) |
| 59 | \( 1 + (0.669 - 0.743i)T \) |
| 61 | \( 1 + (0.669 + 0.743i)T \) |
| 67 | \( 1 + (0.104 + 0.994i)T \) |
| 71 | \( 1 + (-0.104 + 0.994i)T \) |
| 73 | \( 1 + (0.978 - 0.207i)T \) |
| 79 | \( 1 + (-0.104 + 0.994i)T \) |
| 83 | \( 1 + (0.809 - 0.587i)T \) |
| 89 | \( 1 + (0.669 + 0.743i)T \) |
| 97 | \( 1 + (0.104 - 0.994i)T \) |
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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
−23.84362871528587048449586529271, −23.04323308722152147480450215675, −22.3708721815912153730532694562, −21.8230888415272649467068187031, −20.81723990766493824566794021766, −20.081634905778490291882636969526, −19.51578063467229296043573963972, −17.87304517369363420947212119501, −16.92105444105126908489315908377, −16.00200076638353898892018515621, −15.547798719153028075663531557558, −14.74963896073287270511835140661, −13.716524055891536438079928169266, −12.97243109292120058100725947185, −11.95652888012501739033128218061, −10.98984075616677314189011467347, −10.12815284402679042661444735090, −9.15854292692574269503179086654, −8.03431648553082243833889716499, −6.7011162778420175309924456024, −6.00199054209777494057066872254, −4.88347385613331816834818557477, −3.93753614842873599549228917733, −3.293763341033986170458558332123, −2.04542869961561951194952377371,
0.85991506186974565625519968852, 2.22715978186777282856368412893, 3.1559400974263670783298936426, 4.07686751854953397381070703201, 5.6506808717163224348876426169, 6.36043129306460387143368422459, 6.93367806375548565566892047531, 8.266411015704873734781489749320, 9.273463100037306988963399786199, 10.72428567278969050696323264996, 11.43513532163571295848834891678, 12.42752796473686765897628160203, 13.09345975254333028490003793586, 13.83945881646326517884294137756, 14.43657976753868820205180195215, 15.96287825330287801776935030223, 16.21566042363783006013706431620, 17.63691110931348195712977942860, 18.71284584869562436676629076912, 19.392311982729249206154987630490, 20.032858668669711635016465424319, 21.00266202644277821216208606755, 22.0737014388800350281916991981, 22.73945447026127818735222544375, 23.47897943318659212758698860006