L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s − 5-s + 8-s + (0.5 − 0.866i)10-s + 11-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + 23-s + 25-s + (0.5 + 0.866i)26-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s − 5-s + 8-s + (0.5 − 0.866i)10-s + 11-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + 23-s + 25-s + (0.5 + 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.009068532 + 0.1126300311i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.009068532 + 0.1126300311i\) |
\(L(1)\) |
\(\approx\) |
\(0.7721251257 + 0.1745403663i\) |
\(L(1)\) |
\(\approx\) |
\(0.7721251257 + 0.1745403663i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−31.663374751139834458388274007103, −30.76613434959428641946046209356, −29.95905032083069313855805417291, −28.51927555600703961791802260664, −27.76285269568187282247089220934, −26.76427788999370105949447346487, −25.75783157583616088467880602596, −24.11951405073786307700315063247, −22.88302897494825072961120702835, −21.80226343101008922835716030234, −20.54365964799939207609456838974, −19.46621528464890729997444958648, −18.78398123232665122131872388212, −17.262323244442977340349652932302, −16.223381223867594002329689963103, −14.62926047122161226357683094185, −13.066607361733385593164451821340, −11.784747031979044447980676607, −11.075908139655417464021079651530, −9.41140556084223642032906292863, −8.33530949887168361735404759706, −6.92713913626343929795325898470, −4.45056607310803967734795079924, −3.29993053263841558784128284112, −1.208219090186543188809201471520,
0.83327325022601813634542349622, 3.73331291513385510651575210250, 5.36675548432417486792891272818, 6.94083968489108383640188101493, 8.014996026395983674002173075677, 9.21788929630966320384634029624, 10.72877701622394508163292131764, 12.141960796503782292879499281509, 13.88891044009185368615955943932, 15.06239546133681930643429028589, 16.020463689615229918737142828277, 17.090190132324614371653558809257, 18.44498028951957130439477699101, 19.39429148102674447894349654389, 20.5249432869271806590231217374, 22.645981026903088723512508396852, 23.13020106327546438706788430997, 24.57128116719941385609681456635, 25.28726045518376063153938644955, 26.82796670935613978565861191717, 27.39011302290605991398353113837, 28.39582314837269400889392274829, 29.96758891475449991993986197236, 31.28834014941959263804163612217, 32.27234367388550506089539956400