| L(s) = 1 | + 1.34e7·3-s + 6.31e10·5-s − 1.88e13·7-s − 4.35e14·9-s + 5.37e15·11-s + 2.76e17·13-s + 8.51e17·15-s + 6.29e18·17-s − 1.91e18·19-s − 2.53e20·21-s − 1.90e21·23-s − 6.72e20·25-s − 1.42e22·27-s + 5.22e22·29-s + 6.09e22·31-s + 7.25e22·33-s − 1.18e24·35-s − 2.07e24·37-s + 3.73e24·39-s − 5.09e24·41-s − 8.39e24·43-s − 2.74e25·45-s − 2.13e25·47-s + 1.96e26·49-s + 8.49e25·51-s + 1.59e26·53-s + 3.39e26·55-s + ⋯ |
| L(s) = 1 | + 0.542·3-s + 0.924·5-s − 1.49·7-s − 0.705·9-s + 0.388·11-s + 1.49·13-s + 0.502·15-s + 0.533·17-s − 0.0289·19-s − 0.813·21-s − 1.48·23-s − 0.144·25-s − 0.925·27-s + 1.12·29-s + 0.466·31-s + 0.210·33-s − 1.38·35-s − 1.02·37-s + 0.813·39-s − 0.511·41-s − 0.402·43-s − 0.652·45-s − 0.258·47-s + 1.24·49-s + 0.289·51-s + 0.299·53-s + 0.359·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(16)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{33}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 1.34e7T + 6.17e14T^{2} \) |
| 5 | \( 1 - 6.31e10T + 4.65e21T^{2} \) |
| 7 | \( 1 + 1.88e13T + 1.57e26T^{2} \) |
| 11 | \( 1 - 5.37e15T + 1.91e32T^{2} \) |
| 13 | \( 1 - 2.76e17T + 3.40e34T^{2} \) |
| 17 | \( 1 - 6.29e18T + 1.39e38T^{2} \) |
| 19 | \( 1 + 1.91e18T + 4.37e39T^{2} \) |
| 23 | \( 1 + 1.90e21T + 1.63e42T^{2} \) |
| 29 | \( 1 - 5.22e22T + 2.15e45T^{2} \) |
| 31 | \( 1 - 6.09e22T + 1.70e46T^{2} \) |
| 37 | \( 1 + 2.07e24T + 4.11e48T^{2} \) |
| 41 | \( 1 + 5.09e24T + 9.91e49T^{2} \) |
| 43 | \( 1 + 8.39e24T + 4.34e50T^{2} \) |
| 47 | \( 1 + 2.13e25T + 6.83e51T^{2} \) |
| 53 | \( 1 - 1.59e26T + 2.83e53T^{2} \) |
| 59 | \( 1 + 2.16e27T + 7.87e54T^{2} \) |
| 61 | \( 1 + 6.60e27T + 2.21e55T^{2} \) |
| 67 | \( 1 - 2.77e26T + 4.05e56T^{2} \) |
| 71 | \( 1 + 7.68e28T + 2.44e57T^{2} \) |
| 73 | \( 1 - 2.29e28T + 5.79e57T^{2} \) |
| 79 | \( 1 - 2.99e29T + 6.70e58T^{2} \) |
| 83 | \( 1 + 1.89e29T + 3.10e59T^{2} \) |
| 89 | \( 1 + 2.41e30T + 2.69e60T^{2} \) |
| 97 | \( 1 + 3.68e30T + 3.88e61T^{2} \) |
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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
−12.06131541669637342874102915058, −10.33232037578400838041149354325, −9.382570253852432803683094582159, −8.336628187745705255774803512173, −6.44565962062403862600909705905, −5.82787808634420954040231082588, −3.73165511717929586206272876990, −2.85503956122382287994927217842, −1.51836086979972854635709210098, 0,
1.51836086979972854635709210098, 2.85503956122382287994927217842, 3.73165511717929586206272876990, 5.82787808634420954040231082588, 6.44565962062403862600909705905, 8.336628187745705255774803512173, 9.382570253852432803683094582159, 10.33232037578400838041149354325, 12.06131541669637342874102915058
