| L(s) = 1 | + 19.0·2-s + 234.·4-s − 37.6·5-s + 1.15e3·7-s + 2.02e3·8-s − 715.·10-s − 1.61e3·11-s − 7.20e3·13-s + 2.20e4·14-s + 8.48e3·16-s − 2.58e4·17-s − 4.59e4·19-s − 8.80e3·20-s − 3.07e4·22-s + 5.30e4·23-s − 7.67e4·25-s − 1.37e5·26-s + 2.71e5·28-s − 1.64e5·29-s + 6.92e4·31-s − 9.71e4·32-s − 4.91e5·34-s − 4.35e4·35-s − 3.82e5·37-s − 8.75e5·38-s − 7.60e4·40-s + 5.31e5·41-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 1.82·4-s − 0.134·5-s + 1.27·7-s + 1.39·8-s − 0.226·10-s − 0.366·11-s − 0.909·13-s + 2.14·14-s + 0.517·16-s − 1.27·17-s − 1.53·19-s − 0.246·20-s − 0.616·22-s + 0.909·23-s − 0.981·25-s − 1.52·26-s + 2.33·28-s − 1.25·29-s + 0.417·31-s − 0.524·32-s − 2.14·34-s − 0.171·35-s − 1.24·37-s − 2.58·38-s − 0.187·40-s + 1.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 - 19.0T + 128T^{2} \) |
| 5 | \( 1 + 37.6T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.15e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.61e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.20e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.58e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.59e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.30e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.64e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.92e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.82e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.31e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.62e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.23e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.73e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.63e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.66e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.88e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.00e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.62e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.80e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.58e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.69e6T + 8.07e13T^{2} \) |
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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
−9.207659330028578735453428295510, −8.108381979981272780170683158305, −7.20843841413733543852008800960, −6.28663378924193764744130797964, −5.19812710985855383820612626735, −4.64153076990049391718933753762, −3.85959249432967000473662725999, −2.46198419467559134548889482139, −1.88220034610236744903664526886, 0,
1.88220034610236744903664526886, 2.46198419467559134548889482139, 3.85959249432967000473662725999, 4.64153076990049391718933753762, 5.19812710985855383820612626735, 6.28663378924193764744130797964, 7.20843841413733543852008800960, 8.108381979981272780170683158305, 9.207659330028578735453428295510
