| L(s) = 1 | − 15.5i·3-s − 11.1·5-s + 37.6i·7-s − 161.·9-s + 26.6i·11-s − 58.0·13-s + 174. i·15-s + 467.·17-s + 428. i·19-s + 586.·21-s + 360. i·23-s + 125.·25-s + 1.25e3i·27-s − 964.·29-s + 417. i·31-s + ⋯ |
| L(s) = 1 | − 1.73i·3-s − 0.447·5-s + 0.767i·7-s − 1.99·9-s + 0.220i·11-s − 0.343·13-s + 0.774i·15-s + 1.61·17-s + 1.18i·19-s + 1.32·21-s + 0.681i·23-s + 0.200·25-s + 1.72i·27-s − 1.14·29-s + 0.434i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.364393507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.364393507\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 11.1T \) |
| good | 3 | \( 1 + 15.5iT - 81T^{2} \) |
| 7 | \( 1 - 37.6iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 26.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 58.0T + 2.85e4T^{2} \) |
| 17 | \( 1 - 467.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 428. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 360. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 964.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 417. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.79e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 469.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 27.7iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.53e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 276.T + 7.89e6T^{2} \) |
| 59 | \( 1 - 3.81e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.05e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.16e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 5.68e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.00e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 705. iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.62e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 7.15e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.30e4T + 8.85e7T^{2} \) |
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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
−11.48327870094199320655575464029, −10.03672724325856556096967812538, −8.829682718455341470528142249046, −7.78797393938822511993678457198, −7.43616557251622793130273428640, −6.13773091733355195309121242098, −5.40909330011791703892704585657, −3.45330150310350558181476406359, −2.17949946649147224666272206554, −1.07237678539944671606403843971,
0.48887601815441972339256806545, 2.96202293335150580370588758406, 3.92000546175680426972444094592, 4.72567339117138333722635764058, 5.74572766098833285997801898983, 7.27940531353374750370202117743, 8.311473052229278165746788789981, 9.422866422235719990188800707376, 10.01613462219029027826699733224, 10.92699735354875041756550713706
