| L(s) = 1 | − 3.34·2-s − 9·3-s − 20.7·4-s + 30.1·6-s − 42.9·7-s + 176.·8-s + 81·9-s − 121·11-s + 187.·12-s − 1.19e3·13-s + 143.·14-s + 73.9·16-s + 203.·17-s − 271.·18-s + 1.57e3·19-s + 386.·21-s + 405.·22-s − 1.03e3·23-s − 1.59e3·24-s + 4.01e3·26-s − 729·27-s + 893.·28-s + 3.26e3·29-s + 4.56e3·31-s − 5.90e3·32-s + 1.08e3·33-s − 680.·34-s + ⋯ |
| L(s) = 1 | − 0.591·2-s − 0.577·3-s − 0.649·4-s + 0.341·6-s − 0.331·7-s + 0.976·8-s + 0.333·9-s − 0.301·11-s + 0.375·12-s − 1.96·13-s + 0.196·14-s + 0.0721·16-s + 0.170·17-s − 0.197·18-s + 1.00·19-s + 0.191·21-s + 0.178·22-s − 0.408·23-s − 0.563·24-s + 1.16·26-s − 0.192·27-s + 0.215·28-s + 0.720·29-s + 0.852·31-s − 1.01·32-s + 0.174·33-s − 0.101·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 3.34T + 32T^{2} \) |
| 7 | \( 1 + 42.9T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.19e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 203.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.57e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.03e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.56e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.24e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.60e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.64e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.15e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.87e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.93e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.37e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.02e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.60e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.72e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.06e5T + 8.58e9T^{2} \) |
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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
−9.244061328518562005496730754418, −8.169302841343301664416145361528, −7.44683400324082024396417455543, −6.61755648232359531908783541914, −5.19551877491187044582602335169, −4.91061179084178950622188548608, −3.59604962627575371309357256857, −2.25844140098383287200712835959, −0.857772559651459508336897157241, 0,
0.857772559651459508336897157241, 2.25844140098383287200712835959, 3.59604962627575371309357256857, 4.91061179084178950622188548608, 5.19551877491187044582602335169, 6.61755648232359531908783541914, 7.44683400324082024396417455543, 8.169302841343301664416145361528, 9.244061328518562005496730754418
