| L(s) = 1 | + 16·2-s + 256·4-s + 980.·5-s + 9.87e3·7-s + 4.09e3·8-s + 1.56e4·10-s + 8.12e4·11-s + 1.89e5·13-s + 1.57e5·14-s + 6.55e4·16-s − 2.32e5·17-s − 1.35e5·19-s + 2.51e5·20-s + 1.29e6·22-s − 1.28e6·23-s − 9.91e5·25-s + 3.02e6·26-s + 2.52e6·28-s + 1.19e6·29-s + 3.04e6·31-s + 1.04e6·32-s − 3.72e6·34-s + 9.68e6·35-s − 7.74e6·37-s − 2.17e6·38-s + 4.01e6·40-s − 9.13e6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.701·5-s + 1.55·7-s + 0.353·8-s + 0.496·10-s + 1.67·11-s + 1.83·13-s + 1.09·14-s + 0.250·16-s − 0.675·17-s − 0.239·19-s + 0.350·20-s + 1.18·22-s − 0.958·23-s − 0.507·25-s + 1.30·26-s + 0.777·28-s + 0.313·29-s + 0.593·31-s + 0.176·32-s − 0.477·34-s + 1.09·35-s − 0.679·37-s − 0.169·38-s + 0.248·40-s − 0.505·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.905366629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.905366629\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 980.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.87e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.12e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.89e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.32e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.35e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.28e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.19e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.04e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 7.74e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 9.13e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.53e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.78e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.02e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.31e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.57e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 5.69e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.30e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.66e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.26e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.81e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.64e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.17e9T + 7.60e17T^{2} \) |
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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
−11.41064132296071117389270318937, −10.43136419120648518856675866465, −8.976051669608451309983894516086, −8.165179578597050823583719565391, −6.60633006189882885596637256773, −5.89119273022251577938174709862, −4.58575718761834329369687754488, −3.70807643081542247018393820594, −1.87191203005201425688819966662, −1.32794452840952202723729443948,
1.32794452840952202723729443948, 1.87191203005201425688819966662, 3.70807643081542247018393820594, 4.58575718761834329369687754488, 5.89119273022251577938174709862, 6.60633006189882885596637256773, 8.165179578597050823583719565391, 8.976051669608451309983894516086, 10.43136419120648518856675866465, 11.41064132296071117389270318937
