| L(s) = 1 | + 46.7·3-s − 197. i·5-s − 3.79e3i·7-s + 2.18e3·9-s − 1.64e4·11-s − 1.11e4i·13-s − 9.22e3i·15-s + 6.67e4·17-s − 2.04e4·19-s − 1.77e5i·21-s − 4.45e4i·23-s + 3.51e5·25-s + 1.02e5·27-s − 7.25e5i·29-s − 4.42e4i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.315i·5-s − 1.58i·7-s + 0.333·9-s − 1.12·11-s − 0.389i·13-s − 0.182i·15-s + 0.799·17-s − 0.157·19-s − 0.912i·21-s − 0.159i·23-s + 0.900·25-s + 0.192·27-s − 1.02i·29-s − 0.0478i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.538016505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.538016505\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 46.7T \) |
| good | 5 | \( 1 + 197. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 3.79e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.64e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 1.11e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 6.67e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 2.04e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 4.45e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 7.25e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 4.42e4iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 2.85e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 1.07e4T + 7.98e12T^{2} \) |
| 43 | \( 1 - 3.74e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + 2.11e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 5.99e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.11e7T + 1.46e14T^{2} \) |
| 61 | \( 1 - 2.30e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 1.77e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 2.44e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.70e6T + 8.06e14T^{2} \) |
| 79 | \( 1 - 5.98e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 5.83e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 8.20e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 5.81e7T + 7.83e15T^{2} \) |
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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
−9.676629826966762746329173498174, −8.414780981811527604883656699201, −7.70450597881090132300834825834, −7.01158131992061765195843157675, −5.59126143876629943673456824897, −4.50738712218181456818563788880, −3.61174935088914318415300712036, −2.51711550921339800617023228805, −1.12531646387901995197463506978, −0.27416043204052987201991267031,
1.50738330507970349546589034209, 2.62616504901386711267324018692, 3.17731081398473764804276824110, 4.80258080601684775041730726464, 5.62588744658259410144673359105, 6.71876291455356054027313221575, 7.86552898140710330838672576140, 8.596038416401451267196314685553, 9.410842155322407997042139362458, 10.33527596198558355832101326126
