| L(s) = 1 | − 8·2-s + 18.5·3-s + 64·4-s + 6.61·5-s − 148.·6-s + 1.60e3·7-s − 512·8-s − 1.84e3·9-s − 52.9·10-s − 1.17e3·11-s + 1.18e3·12-s − 1.28e4·14-s + 122.·15-s + 4.09e3·16-s + 2.80e4·17-s + 1.47e4·18-s − 1.38e4·19-s + 423.·20-s + 2.97e4·21-s + 9.38e3·22-s − 9.44e4·23-s − 9.47e3·24-s − 7.80e4·25-s − 7.46e4·27-s + 1.02e5·28-s + 1.59e5·29-s − 979.·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.395·3-s + 0.5·4-s + 0.0236·5-s − 0.279·6-s + 1.76·7-s − 0.353·8-s − 0.843·9-s − 0.0167·10-s − 0.265·11-s + 0.197·12-s − 1.25·14-s + 0.00936·15-s + 0.250·16-s + 1.38·17-s + 0.596·18-s − 0.462·19-s + 0.0118·20-s + 0.699·21-s + 0.187·22-s − 1.61·23-s − 0.139·24-s − 0.999·25-s − 0.729·27-s + 0.884·28-s + 1.21·29-s − 0.00662·30-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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 18.5T + 2.18e3T^{2} \) |
| 5 | \( 1 - 6.61T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.60e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.17e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.80e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.38e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.44e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.59e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.13e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.02e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.83e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.85e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.48e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.69e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.33e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 7.19e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.85e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.50e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.07e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.61e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.03e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.14e6T + 8.07e13T^{2} \) |
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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
−9.926660511803081114059887629963, −8.592981264970009912891860252398, −8.192526496412238570792393743975, −7.47799215928709580073437611984, −5.94683240602415251044773561544, −5.06251672401948193431832692607, −3.64152874981948893376892855833, −2.26424254707150264898183908145, −1.45893874660769332962802568287, 0,
1.45893874660769332962802568287, 2.26424254707150264898183908145, 3.64152874981948893376892855833, 5.06251672401948193431832692607, 5.94683240602415251044773561544, 7.47799215928709580073437611984, 8.192526496412238570792393743975, 8.592981264970009912891860252398, 9.926660511803081114059887629963
