| L(s) = 1 | + 4·2-s − 9.47·3-s + 16·4-s + 25·5-s − 37.8·6-s + 37.1·7-s + 64·8-s − 153.·9-s + 100·10-s − 642.·11-s − 151.·12-s + 803.·13-s + 148.·14-s − 236.·15-s + 256·16-s + 285.·17-s − 613.·18-s − 2.35e3·19-s + 400·20-s − 352.·21-s − 2.57e3·22-s − 529·23-s − 606.·24-s + 625·25-s + 3.21e3·26-s + 3.75e3·27-s + 594.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.607·3-s + 0.5·4-s + 0.447·5-s − 0.429·6-s + 0.286·7-s + 0.353·8-s − 0.630·9-s + 0.316·10-s − 1.60·11-s − 0.303·12-s + 1.31·13-s + 0.202·14-s − 0.271·15-s + 0.250·16-s + 0.239·17-s − 0.445·18-s − 1.49·19-s + 0.223·20-s − 0.174·21-s − 1.13·22-s − 0.208·23-s − 0.214·24-s + 0.200·25-s + 0.932·26-s + 0.990·27-s + 0.143·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 4T \) |
| 5 | \( 1 - 25T \) |
| 23 | \( 1 + 529T \) |
| good | 3 | \( 1 + 9.47T + 243T^{2} \) |
| 7 | \( 1 - 37.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 642.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 803.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 285.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.35e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 4.92e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.02e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.34e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.92e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.32e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.64e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.06e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.52e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.75e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.58e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.01e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.05e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.05e5T + 8.58e9T^{2} \) |
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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
−10.91495551574699757688170150044, −10.36865201787981306942771777662, −8.750068715637185044374882517224, −7.79899926446203058446203783921, −6.30822307679827517691332951853, −5.68677348827185313872501447032, −4.71641648741731201090524816590, −3.20401977045866140019417786566, −1.86600890103730857219163369139, 0,
1.86600890103730857219163369139, 3.20401977045866140019417786566, 4.71641648741731201090524816590, 5.68677348827185313872501447032, 6.30822307679827517691332951853, 7.79899926446203058446203783921, 8.750068715637185044374882517224, 10.36865201787981306942771777662, 10.91495551574699757688170150044
