| L(s) = 1 | − 15.7·2-s + 53.8·3-s + 118.·4-s + 331.·5-s − 845.·6-s + 233.·7-s + 143.·8-s + 710.·9-s − 5.21e3·10-s + 6.39e3·12-s − 1.01e4·13-s − 3.66e3·14-s + 1.78e4·15-s − 1.74e4·16-s − 3.11e4·17-s − 1.11e4·18-s − 4.78e4·19-s + 3.94e4·20-s + 1.25e4·21-s − 1.04e4·23-s + 7.72e3·24-s + 3.18e4·25-s + 1.59e5·26-s − 7.94e4·27-s + 2.77e4·28-s + 1.16e5·29-s − 2.80e5·30-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 1.15·3-s + 0.928·4-s + 1.18·5-s − 1.59·6-s + 0.256·7-s + 0.0990·8-s + 0.324·9-s − 1.64·10-s + 1.06·12-s − 1.28·13-s − 0.356·14-s + 1.36·15-s − 1.06·16-s − 1.53·17-s − 0.451·18-s − 1.59·19-s + 1.10·20-s + 0.295·21-s − 0.178·23-s + 0.114·24-s + 0.408·25-s + 1.77·26-s − 0.777·27-s + 0.238·28-s + 0.886·29-s − 1.89·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| good | 2 | \( 1 + 15.7T + 128T^{2} \) |
| 3 | \( 1 - 53.8T + 2.18e3T^{2} \) |
| 5 | \( 1 - 331.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 233.T + 8.23e5T^{2} \) |
| 13 | \( 1 + 1.01e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.11e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.78e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.04e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.16e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.46e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.27e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.51e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.28e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.80e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.47e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.03e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 4.22e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.11e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.10e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 7.83e4T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.59e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.52e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.87e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.19e6T + 8.07e13T^{2} \) |
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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
−11.05805682011905855085968359775, −10.09984266003985480432085599608, −9.248743358893016039251781664486, −8.678184917005123681003430961644, −7.63587544203999139694108351267, −6.41565911540366600557401545749, −4.57977345868339494316531625560, −2.41209008975458252547648451861, −1.91209192988283299633393324318, 0,
1.91209192988283299633393324318, 2.41209008975458252547648451861, 4.57977345868339494316531625560, 6.41565911540366600557401545749, 7.63587544203999139694108351267, 8.678184917005123681003430961644, 9.248743358893016039251781664486, 10.09984266003985480432085599608, 11.05805682011905855085968359775
