| L(s) = 1 | − 2·2-s + 0.791·3-s + 4·4-s + 1.75·5-s − 1.58·6-s + 7·7-s − 8·8-s − 26.3·9-s − 3.51·10-s + 30.9·11-s + 3.16·12-s − 52.8·13-s − 14·14-s + 1.39·15-s + 16·16-s − 121.·17-s + 52.7·18-s + 134.·19-s + 7.03·20-s + 5.54·21-s − 61.9·22-s − 23·23-s − 6.33·24-s − 121.·25-s + 105.·26-s − 42.2·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.152·3-s + 0.5·4-s + 0.157·5-s − 0.107·6-s + 0.377·7-s − 0.353·8-s − 0.976·9-s − 0.111·10-s + 0.848·11-s + 0.0761·12-s − 1.12·13-s − 0.267·14-s + 0.0239·15-s + 0.250·16-s − 1.73·17-s + 0.690·18-s + 1.62·19-s + 0.0786·20-s + 0.0575·21-s − 0.599·22-s − 0.208·23-s − 0.0538·24-s − 0.975·25-s + 0.797·26-s − 0.301·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 0.791T + 27T^{2} \) |
| 5 | \( 1 - 1.75T + 125T^{2} \) |
| 11 | \( 1 - 30.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 134.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 36.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 103.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 386.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 19.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 125.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 181.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 304.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 715.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 737.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 732.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 182.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 53.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 242.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 896.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 229.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 197.T + 9.12e5T^{2} \) |
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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
−10.70953467176935239721855008364, −9.546394182203310603565112874041, −8.990956512150100131037670089489, −7.962856101907618813340521377076, −7.01399546776554016804806687421, −5.92471413427961390885876460161, −4.67293851968786216673760984665, −3.04716099653795769272071637181, −1.79485228194279366197250753080, 0,
1.79485228194279366197250753080, 3.04716099653795769272071637181, 4.67293851968786216673760984665, 5.92471413427961390885876460161, 7.01399546776554016804806687421, 7.962856101907618813340521377076, 8.990956512150100131037670089489, 9.546394182203310603565112874041, 10.70953467176935239721855008364
