| L(s) = 1 | − 2.75i·3-s + 3.84i·7-s + 1.41·9-s + 6.19i·11-s − 16.1·13-s + 5.20·17-s − 36.2i·19-s + 10.6·21-s − 22.0i·23-s − 28.6i·27-s + 20.0·29-s − 26.4i·31-s + 17.0·33-s − 69.3·37-s + 44.3i·39-s + ⋯ |
| L(s) = 1 | − 0.918i·3-s + 0.549i·7-s + 0.157·9-s + 0.562i·11-s − 1.23·13-s + 0.306·17-s − 1.90i·19-s + 0.504·21-s − 0.958i·23-s − 1.06i·27-s + 0.690·29-s − 0.852i·31-s + 0.516·33-s − 1.87·37-s + 1.13i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.242549764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242549764\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.75iT - 9T^{2} \) |
| 7 | \( 1 - 3.84iT - 49T^{2} \) |
| 11 | \( 1 - 6.19iT - 121T^{2} \) |
| 13 | \( 1 + 16.1T + 169T^{2} \) |
| 17 | \( 1 - 5.20T + 289T^{2} \) |
| 19 | \( 1 + 36.2iT - 361T^{2} \) |
| 23 | \( 1 + 22.0iT - 529T^{2} \) |
| 29 | \( 1 - 20.0T + 841T^{2} \) |
| 31 | \( 1 + 26.4iT - 961T^{2} \) |
| 37 | \( 1 + 69.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 11.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 25.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 66.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 39.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 27.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 54.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 107. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 70.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 37.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 97.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 133.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 6.40T + 9.40e3T^{2} \) |
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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
−9.699693280688634766516321150068, −8.876567426530972115321959461714, −7.888988762780026231312017549138, −7.05829987350494190048104828280, −6.56820948721952969774289273168, −5.23135024699211183684087172559, −4.46725126321120811187664346524, −2.77276896437911038209536019869, −1.98964158820306174654353114428, −0.41956301212726522335579867376,
1.47850759151870497533510647042, 3.18459328809804150312643511311, 3.97889698292791341097112480493, 4.94943783954622096805061227994, 5.79231145230807183000431273751, 7.06498076155906971414189099480, 7.78528713572693112465739858984, 8.829109024482082040755588852550, 9.802366224867313659446654752851, 10.24209782357494541842578031576
