| L(s) = 1 | − 15.6·3-s − 33.5·5-s − 178.·7-s + 3.41·9-s + 240.·11-s − 1.05e3·13-s + 527.·15-s − 2.04e3·17-s − 692.·19-s + 2.79e3·21-s − 3.73e3·23-s − 1.99e3·25-s + 3.76e3·27-s + 1.84e3·29-s + 3.21e3·31-s − 3.78e3·33-s + 5.98e3·35-s − 3.39e3·37-s + 1.65e4·39-s − 5.96e3·41-s − 770.·43-s − 114.·45-s − 1.70e4·47-s + 1.48e4·49-s + 3.21e4·51-s − 3.21e4·53-s − 8.09e3·55-s + ⋯ |
| L(s) = 1 | − 1.00·3-s − 0.600·5-s − 1.37·7-s + 0.0140·9-s + 0.600·11-s − 1.73·13-s + 0.605·15-s − 1.71·17-s − 0.440·19-s + 1.38·21-s − 1.47·23-s − 0.638·25-s + 0.992·27-s + 0.407·29-s + 0.600·31-s − 0.604·33-s + 0.825·35-s − 0.407·37-s + 1.74·39-s − 0.553·41-s − 0.0635·43-s − 0.00845·45-s − 1.12·47-s + 0.885·49-s + 1.73·51-s − 1.57·53-s − 0.360·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.03809828788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03809828788\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 15.6T + 243T^{2} \) |
| 5 | \( 1 + 33.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 178.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 240.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.05e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.04e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 692.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.73e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.84e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.21e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.39e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.96e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 770.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.70e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.21e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.40e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.36e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.88e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.54e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.26e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.77e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.65e4T + 8.58e9T^{2} \) |
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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
−11.36033992228152713230838875704, −10.19669829078150833384096303084, −9.470235127698191572412096504034, −8.194099497777264511815057775505, −6.77189273836201075480854495162, −6.37629310566805637497799379617, −4.98352601014970609226503447023, −3.90169909912820454203357769925, −2.41434838430341572621818949912, −0.11034757927099735899535689951,
0.11034757927099735899535689951, 2.41434838430341572621818949912, 3.90169909912820454203357769925, 4.98352601014970609226503447023, 6.37629310566805637497799379617, 6.77189273836201075480854495162, 8.194099497777264511815057775505, 9.470235127698191572412096504034, 10.19669829078150833384096303084, 11.36033992228152713230838875704
