| L(s) = 1 | + 16.0·2-s + 27·3-s + 128.·4-s + 322.·5-s + 432.·6-s − 992.·7-s + 15.6·8-s + 729·9-s + 5.17e3·10-s + 3.48e3·12-s − 1.44e4·13-s − 1.59e4·14-s + 8.71e3·15-s − 1.62e4·16-s + 2.12e4·17-s + 1.16e4·18-s − 4.37e4·19-s + 4.16e4·20-s − 2.68e4·21-s + 2.25e4·23-s + 423.·24-s + 2.59e4·25-s − 2.31e5·26-s + 1.96e4·27-s − 1.28e5·28-s − 1.80e5·29-s + 1.39e5·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 1.00·4-s + 1.15·5-s + 0.818·6-s − 1.09·7-s + 0.0108·8-s + 0.333·9-s + 1.63·10-s + 0.581·12-s − 1.82·13-s − 1.55·14-s + 0.666·15-s − 0.992·16-s + 1.04·17-s + 0.472·18-s − 1.46·19-s + 1.16·20-s − 0.631·21-s + 0.386·23-s + 0.00625·24-s + 0.332·25-s − 2.58·26-s + 0.192·27-s − 1.10·28-s − 1.37·29-s + 0.944·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 27T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 16.0T + 128T^{2} \) |
| 5 | \( 1 - 322.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 992.T + 8.23e5T^{2} \) |
| 13 | \( 1 + 1.44e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.12e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.37e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.25e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.80e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.33e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.68e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.00e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.66e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.82e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.20e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.35e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.07e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.42e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.43e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.49e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.08e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.67e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.91e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.41e6T + 8.07e13T^{2} \) |
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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
−9.604955655780244291007432718284, −9.268845886918186809959344202718, −7.58522102919080664923083369365, −6.56584643559359575445464176369, −5.78655392668779985815996019556, −4.88003891533934299214751980546, −3.71467282670715010753260989366, −2.73354421418563840616816095944, −2.02926686267831389540881537539, 0,
2.02926686267831389540881537539, 2.73354421418563840616816095944, 3.71467282670715010753260989366, 4.88003891533934299214751980546, 5.78655392668779985815996019556, 6.56584643559359575445464176369, 7.58522102919080664923083369365, 9.268845886918186809959344202718, 9.604955655780244291007432718284
