| L(s) = 1 | − 4.82i·3-s + 7.65·5-s − 1.65i·7-s − 14.3·9-s + 1.51i·11-s − 0.343·13-s − 36.9i·15-s − 13.3·17-s − 20.8i·19-s − 7.99·21-s + 33.6i·23-s + 33.6·25-s + 25.6i·27-s + 39.6·29-s + 45.2i·31-s + ⋯ |
| L(s) = 1 | − 1.60i·3-s + 1.53·5-s − 0.236i·7-s − 1.59·9-s + 0.137i·11-s − 0.0263·13-s − 2.46i·15-s − 0.783·17-s − 1.09i·19-s − 0.380·21-s + 1.46i·23-s + 1.34·25-s + 0.950i·27-s + 1.36·29-s + 1.45i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.18105 - 1.18105i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18105 - 1.18105i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 4.82iT - 9T^{2} \) |
| 5 | \( 1 - 7.65T + 25T^{2} \) |
| 7 | \( 1 + 1.65iT - 49T^{2} \) |
| 11 | \( 1 - 1.51iT - 121T^{2} \) |
| 13 | \( 1 + 0.343T + 169T^{2} \) |
| 17 | \( 1 + 13.3T + 289T^{2} \) |
| 19 | \( 1 + 20.8iT - 361T^{2} \) |
| 23 | \( 1 - 33.6iT - 529T^{2} \) |
| 29 | \( 1 - 39.6T + 841T^{2} \) |
| 31 | \( 1 - 45.2iT - 961T^{2} \) |
| 37 | \( 1 + 29.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 24.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 50.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 16.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 53.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 62.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 40.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 55.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 137. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 114. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 2.56T + 7.92e3T^{2} \) |
| 97 | \( 1 + 138.T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.12250915849331709151576548753, −12.10316647018077563993494964814, −10.88148552984951934479646734496, −9.601980717139849496589579706215, −8.519573119252287733232986077623, −7.09489209188696859639666775396, −6.45176504491950346507659847280, −5.23995189278967947793786705523, −2.57969419433169744076197949795, −1.36661546252875641113770709066,
2.49725236841593127540615622618, 4.22299301065594092408425940155, 5.40808767942685456439004065765, 6.33925023231608716390854716029, 8.534999803588251891650414199663, 9.396247334220179361214100043013, 10.20358180945590774240492582205, 10.81555378577993258648967205929, 12.31404907078912960660543870578, 13.64651021471369535263617511127
