| L(s) = 1 | − 1.73i·3-s − 2.63·5-s − 12.5i·7-s − 2.99·9-s + 5.79i·11-s + 8.78·13-s + 4.56i·15-s − 30.1·17-s + 17.4i·19-s − 21.7·21-s − 2.48i·23-s − 18.0·25-s + 5.19i·27-s − 26.4·29-s − 38.0i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.527·5-s − 1.79i·7-s − 0.333·9-s + 0.527i·11-s + 0.675·13-s + 0.304i·15-s − 1.77·17-s + 0.916i·19-s − 1.03·21-s − 0.108i·23-s − 0.722·25-s + 0.192i·27-s − 0.910·29-s − 1.22i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(-0.652909i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.652909i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| good | 5 | \( 1 + 2.63T + 25T^{2} \) |
| 7 | \( 1 + 12.5iT - 49T^{2} \) |
| 11 | \( 1 - 5.79iT - 121T^{2} \) |
| 13 | \( 1 - 8.78T + 169T^{2} \) |
| 17 | \( 1 + 30.1T + 289T^{2} \) |
| 19 | \( 1 - 17.4iT - 361T^{2} \) |
| 23 | \( 1 + 2.48iT - 529T^{2} \) |
| 29 | \( 1 + 26.4T + 841T^{2} \) |
| 31 | \( 1 + 38.0iT - 961T^{2} \) |
| 37 | \( 1 + 47.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 53.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 30.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 16.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 49.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 107. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 62.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 60.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 19.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.13T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.83iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 159. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 39.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 60.5T + 9.40e3T^{2} \) |
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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
−10.84703568816806946827878843289, −9.879537584911404454054968659703, −8.647201796808442155269668191195, −7.61207866941587272808773619014, −7.10866050496142840262921912926, −6.05751341696266129524685321111, −4.37972897683909811264559094139, −3.74253956451734473236375963582, −1.82764263738140355806353544754, −0.27658940651158177774747829322,
2.26585267980616344166614683539, 3.46537087984794372110121460785, 4.75614335748711971211001773386, 5.72877586048080339223002369783, 6.68510961616099132745238525124, 8.240084623802408321019220381654, 8.856923333482471708025142905506, 9.471660293370832467401520846306, 11.13250236557800454386556233634, 11.22316317610918181309317963996
