| L(s) = 1 | + 15.5·2-s + 113.·4-s + 59.4·5-s − 1.11e3·7-s − 230.·8-s + 923.·10-s + 1.33e3·11-s − 2.64e3·13-s − 1.73e4·14-s − 1.80e4·16-s − 1.87e4·17-s − 3.97e4·19-s + 6.72e3·20-s + 2.06e4·22-s + 4.97e4·23-s − 7.45e4·25-s − 4.10e4·26-s − 1.26e5·28-s + 1.52e5·29-s − 9.22e4·31-s − 2.51e5·32-s − 2.90e5·34-s − 6.63e4·35-s + 2.17e5·37-s − 6.16e5·38-s − 1.36e4·40-s − 6.71e5·41-s + ⋯ |
| L(s) = 1 | + 1.37·2-s + 0.884·4-s + 0.212·5-s − 1.22·7-s − 0.158·8-s + 0.292·10-s + 0.301·11-s − 0.333·13-s − 1.68·14-s − 1.10·16-s − 0.924·17-s − 1.32·19-s + 0.188·20-s + 0.413·22-s + 0.852·23-s − 0.954·25-s − 0.458·26-s − 1.08·28-s + 1.16·29-s − 0.556·31-s − 1.35·32-s − 1.26·34-s − 0.261·35-s + 0.706·37-s − 1.82·38-s − 0.0338·40-s − 1.52·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - 1.33e3T \) |
| good | 2 | \( 1 - 15.5T + 128T^{2} \) |
| 5 | \( 1 - 59.4T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.11e3T + 8.23e5T^{2} \) |
| 13 | \( 1 + 2.64e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.87e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.97e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.97e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.52e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.22e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.17e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.71e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.29e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.38e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.76e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.23e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.93e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 9.30e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.28e4T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.18e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.90e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.48e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.84e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.59e7T + 8.07e13T^{2} \) |
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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
−12.41722510370665211487495640390, −11.22103069179480955156621805194, −9.864730087102120123617491203307, −8.769857774649066134478688159781, −6.80150445011899698238541626204, −6.14592053611494633846547659671, −4.74886502587136817594660493809, −3.60414436658870993865416699742, −2.38991629883758509442825735845, 0,
2.38991629883758509442825735845, 3.60414436658870993865416699742, 4.74886502587136817594660493809, 6.14592053611494633846547659671, 6.80150445011899698238541626204, 8.769857774649066134478688159781, 9.864730087102120123617491203307, 11.22103069179480955156621805194, 12.41722510370665211487495640390
