| L(s) = 1 | + 8·2-s + 2.83·3-s + 64·4-s + 91.0·5-s + 22.6·6-s − 1.50e3·7-s + 512·8-s − 2.17e3·9-s + 728.·10-s − 4.28e3·11-s + 181.·12-s + 5.21e3·13-s − 1.20e4·14-s + 257.·15-s + 4.09e3·16-s + 1.42e4·17-s − 1.74e4·18-s − 3.23e4·19-s + 5.82e3·20-s − 4.26e3·21-s − 3.42e4·22-s − 5.80e4·23-s + 1.44e3·24-s − 6.98e4·25-s + 4.16e4·26-s − 1.23e4·27-s − 9.63e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0605·3-s + 0.5·4-s + 0.325·5-s + 0.0428·6-s − 1.65·7-s + 0.353·8-s − 0.996·9-s + 0.230·10-s − 0.970·11-s + 0.0302·12-s + 0.657·13-s − 1.17·14-s + 0.0197·15-s + 0.250·16-s + 0.702·17-s − 0.704·18-s − 1.08·19-s + 0.162·20-s − 0.100·21-s − 0.686·22-s − 0.994·23-s + 0.0214·24-s − 0.893·25-s + 0.465·26-s − 0.120·27-s − 0.829·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 8T \) |
| 31 | \( 1 + 2.97e4T \) |
| good | 3 | \( 1 - 2.83T + 2.18e3T^{2} \) |
| 5 | \( 1 - 91.0T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.50e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.28e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.21e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.42e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.23e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.80e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.78e3T + 1.72e10T^{2} \) |
| 37 | \( 1 - 3.73e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.91e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.13e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.01e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.15e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.67e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.77e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.94e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 9.83e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 8.21e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.64e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.62e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.79e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.44e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.13413966196194037997380613300, −12.12976304928495909902689560599, −10.70158768954350124296565156396, −9.641727747429270511364547817972, −8.116226551732932223150186348770, −6.40488445829395452981362573660, −5.65007154123486545401861352316, −3.66957817129584531728743901546, −2.51223780631892716088860619157, 0,
2.51223780631892716088860619157, 3.66957817129584531728743901546, 5.65007154123486545401861352316, 6.40488445829395452981362573660, 8.116226551732932223150186348770, 9.641727747429270511364547817972, 10.70158768954350124296565156396, 12.12976304928495909902689560599, 13.13413966196194037997380613300
