| L(s) = 1 | − 46.7i·3-s − 19.7·5-s − 759. i·7-s − 2.18e3·9-s + 1.58e4i·11-s − 4.76e4·13-s + 923. i·15-s − 1.17e5·17-s − 1.96e5i·19-s − 3.55e4·21-s + 1.43e5i·23-s − 3.90e5·25-s + 1.02e5i·27-s + 1.30e6·29-s − 1.08e6i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.0315·5-s − 0.316i·7-s − 0.333·9-s + 1.08i·11-s − 1.66·13-s + 0.0182i·15-s − 1.41·17-s − 1.51i·19-s − 0.182·21-s + 0.513i·23-s − 0.999·25-s + 0.192i·27-s + 1.84·29-s − 1.17i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.129053195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.129053195\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 46.7iT \) |
| good | 5 | \( 1 + 19.7T + 3.90e5T^{2} \) |
| 7 | \( 1 + 759. iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.58e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.76e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.17e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.96e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.43e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 1.30e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.08e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 2.61e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 9.56e5T + 7.98e12T^{2} \) |
| 43 | \( 1 - 4.78e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 2.03e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 4.50e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.40e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.05e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.77e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.22e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.34e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 1.08e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 7.99e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 7.46e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 5.73e7T + 7.83e15T^{2} \) |
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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
−9.856843543114831784405405110729, −9.182587005427997943389614561448, −7.916481748332689601538763468709, −7.14760538558271972858106990434, −6.51702317835639966462571621869, −5.00156649256484248051414205829, −4.36091195501631628066751711968, −2.67157139634169891182588496051, −2.00728624825848574670529423492, −0.53951011517732182278612083871,
0.36594003545279591378081260435, 2.02045540286486815229657534268, 3.02937781829100364436430389741, 4.17743586394322889708271098069, 5.13887609907131853146917627724, 6.08159001529457903098920421420, 7.17124314907220805004146105466, 8.395595409009706522487892026223, 8.959457032683983347523037111804, 10.18111564211508805250025442167
