| L(s) = 1 | − 9.71·2-s + 14.3·3-s + 62.3·4-s − 25·5-s − 139.·6-s − 67.8·7-s − 294.·8-s − 37.0·9-s + 242.·10-s + 121·11-s + 894.·12-s − 692.·13-s + 658.·14-s − 358.·15-s + 866.·16-s − 1.45e3·17-s + 359.·18-s − 361·19-s − 1.55e3·20-s − 973.·21-s − 1.17e3·22-s − 2.58e3·23-s − 4.22e3·24-s + 625·25-s + 6.72e3·26-s − 4.01e3·27-s − 4.22e3·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 0.920·3-s + 1.94·4-s − 0.447·5-s − 1.58·6-s − 0.523·7-s − 1.62·8-s − 0.152·9-s + 0.767·10-s + 0.301·11-s + 1.79·12-s − 1.13·13-s + 0.898·14-s − 0.411·15-s + 0.846·16-s − 1.22·17-s + 0.261·18-s − 0.229·19-s − 0.871·20-s − 0.481·21-s − 0.517·22-s − 1.01·23-s − 1.49·24-s + 0.200·25-s + 1.95·26-s − 1.06·27-s − 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2192767822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2192767822\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 9.71T + 32T^{2} \) |
| 3 | \( 1 - 14.3T + 243T^{2} \) |
| 7 | \( 1 + 67.8T + 1.68e4T^{2} \) |
| 13 | \( 1 + 692.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.45e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.58e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.89e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 991.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 302.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.58e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.00e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.38e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.15e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.54e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.56e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.41e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.58e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.08e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.87e4T + 8.58e9T^{2} \) |
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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
−9.153560535068319397895505675090, −8.339894144027791196378768161375, −7.994922293517533825539741357024, −6.94175710345335015341103270692, −6.41671084290273915833652536668, −4.76767198531021015599036877453, −3.46687941764919102409344493610, −2.52392273428739625480496901738, −1.77220508285875427473878822978, −0.23635537100412573926347223559,
0.23635537100412573926347223559, 1.77220508285875427473878822978, 2.52392273428739625480496901738, 3.46687941764919102409344493610, 4.76767198531021015599036877453, 6.41671084290273915833652536668, 6.94175710345335015341103270692, 7.994922293517533825539741357024, 8.339894144027791196378768161375, 9.153560535068319397895505675090
