| L(s) = 1 | + 149.·5-s − 3.20e3i·7-s − 3.23e3i·11-s + 176.·13-s − 8.04e4·17-s + 2.39e5i·19-s + 8.53e4i·23-s − 3.68e5·25-s + 3.10e5·29-s + 4.94e5i·31-s − 4.78e5i·35-s − 8.55e5·37-s + 1.41e6·41-s + 2.15e6i·43-s − 1.10e6i·47-s + ⋯ |
| L(s) = 1 | + 0.238·5-s − 1.33i·7-s − 0.220i·11-s + 0.00618·13-s − 0.963·17-s + 1.83i·19-s + 0.304i·23-s − 0.942·25-s + 0.438·29-s + 0.535i·31-s − 0.319i·35-s − 0.456·37-s + 0.499·41-s + 0.629i·43-s − 0.226i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.643507951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.643507951\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 149.T + 3.90e5T^{2} \) |
| 7 | \( 1 + 3.20e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 3.23e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 176.T + 8.15e8T^{2} \) |
| 17 | \( 1 + 8.04e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 2.39e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 8.53e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 3.10e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 4.94e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 8.55e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.41e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 2.15e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 1.10e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.47e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.62e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 9.65e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.25e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 3.04e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.01e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 4.96e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 4.44e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 1.11e8T + 3.93e15T^{2} \) |
| 97 | \( 1 - 3.96e6T + 7.83e15T^{2} \) |
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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
−10.41576422320604478179117675041, −9.815692504531880083989935495137, −8.536806631089551114125182616486, −7.61901687478760654262013110116, −6.68020367343951824520532522931, −5.64288428218627371750380938840, −4.30892083624906074124011971257, −3.52745318814347869611500199848, −1.96518909676485849594635824418, −0.869743100232294776089753151502,
0.40516204048829246010762115070, 2.06065702940071154646389393557, 2.72914652567451870565429989555, 4.32973265871658284778313091982, 5.35907698237912824349684123174, 6.29664930667967408049737196132, 7.32014907840648630481640017470, 8.720309819982887331885737298808, 9.094029806342980055484740494083, 10.24840653328332157591745518510
