| L(s) = 1 | + 1.93·2-s + 7.20·3-s − 4.26·4-s − 5·5-s + 13.9·6-s − 6.82·7-s − 23.6·8-s + 24.9·9-s − 9.65·10-s + 11·11-s − 30.7·12-s + 86.8·13-s − 13.1·14-s − 36.0·15-s − 11.6·16-s − 66.7·17-s + 48.1·18-s − 19·19-s + 21.3·20-s − 49.1·21-s + 21.2·22-s − 189.·23-s − 170.·24-s + 25·25-s + 167.·26-s − 15.0·27-s + 29.1·28-s + ⋯ |
| L(s) = 1 | + 0.682·2-s + 1.38·3-s − 0.533·4-s − 0.447·5-s + 0.946·6-s − 0.368·7-s − 1.04·8-s + 0.922·9-s − 0.305·10-s + 0.301·11-s − 0.739·12-s + 1.85·13-s − 0.251·14-s − 0.620·15-s − 0.181·16-s − 0.952·17-s + 0.630·18-s − 0.229·19-s + 0.238·20-s − 0.510·21-s + 0.205·22-s − 1.71·23-s − 1.45·24-s + 0.200·25-s + 1.26·26-s − 0.107·27-s + 0.196·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 1.93T + 8T^{2} \) |
| 3 | \( 1 - 7.20T + 27T^{2} \) |
| 7 | \( 1 + 6.82T + 343T^{2} \) |
| 13 | \( 1 - 86.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 66.7T + 4.91e3T^{2} \) |
| 23 | \( 1 + 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 113.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 232.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 364.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 57.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 77.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 121.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 783.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 398.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 513.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 302.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 550.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 101.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 959.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.21e3T + 9.12e5T^{2} \) |
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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
−8.937621614699623107583985520821, −8.500839205536298508755480910118, −7.68574478308425112925916128684, −6.45107976815579037572119668460, −5.69892411960771241673774903390, −4.16683937809803427475273282954, −3.86298356248107538282222407837, −3.05306602923255548163481716526, −1.76435432111447108379068400969, 0,
1.76435432111447108379068400969, 3.05306602923255548163481716526, 3.86298356248107538282222407837, 4.16683937809803427475273282954, 5.69892411960771241673774903390, 6.45107976815579037572119668460, 7.68574478308425112925916128684, 8.500839205536298508755480910118, 8.937621614699623107583985520821
