| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s − 140.·5-s + 216·6-s − 1.60e3·7-s − 512·8-s + 729·9-s + 1.12e3·10-s − 916.·11-s − 1.72e3·12-s − 1.46e4·13-s + 1.28e4·14-s + 3.78e3·15-s + 4.09e3·16-s + 2.76e4·17-s − 5.83e3·18-s + 2.50e4·19-s − 8.96e3·20-s + 4.33e4·21-s + 7.32e3·22-s + 7.64e4·23-s + 1.38e4·24-s − 5.85e4·25-s + 1.17e5·26-s − 1.96e4·27-s − 1.02e5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.500·5-s + 0.408·6-s − 1.76·7-s − 0.353·8-s + 0.333·9-s + 0.354·10-s − 0.207·11-s − 0.288·12-s − 1.85·13-s + 1.25·14-s + 0.289·15-s + 0.250·16-s + 1.36·17-s − 0.235·18-s + 0.837·19-s − 0.250·20-s + 1.02·21-s + 0.146·22-s + 1.30·23-s + 0.204·24-s − 0.749·25-s + 1.31·26-s − 0.192·27-s − 0.884·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 8T \) |
| 3 | \( 1 + 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 5 | \( 1 + 140.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.60e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 916.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.46e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.76e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.50e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.64e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 5.19e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.18e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.13e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.93e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.36e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.86e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.86e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.30e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.01e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.19e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.49e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.66e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.55e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.17e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.72e7T + 8.07e13T^{2} \) |
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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
−9.772887221009773878082356703933, −9.253080298817990759592067865321, −7.59771914376907422570284043098, −7.22191470489468241604816527426, −6.09272270188075462536197572355, −5.08543044313293863655714883979, −3.51601406692078436612791391305, −2.63369399844348757747650038181, −0.826403739252870679868788116140, 0,
0.826403739252870679868788116140, 2.63369399844348757747650038181, 3.51601406692078436612791391305, 5.08543044313293863655714883979, 6.09272270188075462536197572355, 7.22191470489468241604816527426, 7.59771914376907422570284043098, 9.253080298817990759592067865321, 9.772887221009773878082356703933
