| L(s) = 1 | − 8·2-s − 39.9·3-s + 64·4-s − 323.·5-s + 319.·6-s + 568.·7-s − 512·8-s − 588.·9-s + 2.58e3·10-s − 238.·11-s − 2.55e3·12-s − 4.54e3·14-s + 1.29e4·15-s + 4.09e3·16-s + 2.04e4·17-s + 4.70e3·18-s − 9.64e3·19-s − 2.07e4·20-s − 2.27e4·21-s + 1.90e3·22-s − 7.82e4·23-s + 2.04e4·24-s + 2.66e4·25-s + 1.10e5·27-s + 3.63e4·28-s − 1.38e5·29-s − 1.03e5·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.855·3-s + 0.5·4-s − 1.15·5-s + 0.604·6-s + 0.626·7-s − 0.353·8-s − 0.268·9-s + 0.818·10-s − 0.0539·11-s − 0.427·12-s − 0.443·14-s + 0.990·15-s + 0.250·16-s + 1.01·17-s + 0.190·18-s − 0.322·19-s − 0.579·20-s − 0.535·21-s + 0.0381·22-s − 1.34·23-s + 0.302·24-s + 0.341·25-s + 1.08·27-s + 0.313·28-s − 1.05·29-s − 0.700·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3492509164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3492509164\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 39.9T + 2.18e3T^{2} \) |
| 5 | \( 1 + 323.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 568.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 238.T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.04e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 9.64e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.82e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.38e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.60e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.52e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.85e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.50e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.20e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.65e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.48e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.04e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.87e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.68e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.57e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.29e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.93e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.15e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.33e6T + 8.07e13T^{2} \) |
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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
−10.55872724301638611268295221969, −9.456033705737520298846435036877, −8.055269230927260252564143723358, −7.926980555370721384110610346743, −6.56363104845749508210945193692, −5.57867484772998308280544509253, −4.44324204928085912764662509741, −3.20527214781971589569875194324, −1.60443714002249210641926510294, −0.33240100619318930520040079901,
0.33240100619318930520040079901, 1.60443714002249210641926510294, 3.20527214781971589569875194324, 4.44324204928085912764662509741, 5.57867484772998308280544509253, 6.56363104845749508210945193692, 7.926980555370721384110610346743, 8.055269230927260252564143723358, 9.456033705737520298846435036877, 10.55872724301638611268295221969
