| L(s) = 1 | + 31.8i·2-s − 758.·4-s + 1.91e3·7-s − 1.60e4i·8-s − 1.81e4i·11-s − 1.87e4·13-s + 6.10e4i·14-s + 3.15e5·16-s − 8.30e4i·17-s + 3.32e4·19-s + 5.79e5·22-s − 1.14e5i·23-s − 5.96e5i·26-s − 1.45e6·28-s + 7.60e5i·29-s + ⋯ |
| L(s) = 1 | + 1.99i·2-s − 2.96·4-s + 0.798·7-s − 3.90i·8-s − 1.24i·11-s − 0.655·13-s + 1.58i·14-s + 4.81·16-s − 0.993i·17-s + 0.254·19-s + 2.47·22-s − 0.410i·23-s − 1.30i·26-s − 2.36·28-s + 1.07i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.7268201227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7268201227\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 31.8iT - 256T^{2} \) |
| 7 | \( 1 - 1.91e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.81e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 1.87e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 8.30e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 3.32e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 1.14e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 7.60e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 8.34e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.03e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 5.38e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 3.56e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 1.25e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 6.58e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 8.31e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.71e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 9.49e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 2.26e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.85e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.77e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + 5.83e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 2.20e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 6.94e7T + 7.83e15T^{2} \) |
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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
−11.44400809325721310589262833716, −10.02723012772066769722778981669, −8.990300371714708496376295948853, −8.268775075725048388284225775741, −7.41518514254978715841571715711, −6.50735998019419976299412478067, −5.35867070969925306165262066630, −4.79120222279556847368087362499, −3.38882900086221834010475290911, −0.956259383947528105566919167283,
0.20002310269687040782679068270, 1.64763530351394574766222429756, 2.16702833255920544436528790409, 3.61826638169239401013612107126, 4.54869282521349195862800524739, 5.40220180829392690283019328603, 7.56863880064018663119895854375, 8.580649719658786834729503669989, 9.605589010114312935228599229715, 10.28749476912988517771168219921
