| L(s) = 1 | + 3.36·2-s − 3·3-s + 3.33·4-s − 9.34·5-s − 10.0·6-s − 20.4·7-s − 15.7·8-s + 9·9-s − 31.4·10-s − 31.9·11-s − 10.0·12-s + 71.3·13-s − 68.7·14-s + 28.0·15-s − 79.5·16-s − 79.2·17-s + 30.2·18-s − 31.1·20-s + 61.2·21-s − 107.·22-s − 90.4·23-s + 47.1·24-s − 37.6·25-s + 240.·26-s − 27·27-s − 68.0·28-s + 20.9·29-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 0.577·3-s + 0.416·4-s − 0.836·5-s − 0.687·6-s − 1.10·7-s − 0.694·8-s + 0.333·9-s − 0.995·10-s − 0.877·11-s − 0.240·12-s + 1.52·13-s − 1.31·14-s + 0.482·15-s − 1.24·16-s − 1.13·17-s + 0.396·18-s − 0.348·20-s + 0.636·21-s − 1.04·22-s − 0.820·23-s + 0.400·24-s − 0.300·25-s + 1.81·26-s − 0.192·27-s − 0.459·28-s + 0.133·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.086612040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.086612040\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 3.36T + 8T^{2} \) |
| 5 | \( 1 + 9.34T + 125T^{2} \) |
| 7 | \( 1 + 20.4T + 343T^{2} \) |
| 11 | \( 1 + 31.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 71.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 79.2T + 4.91e3T^{2} \) |
| 23 | \( 1 + 90.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 134.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 235.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 322.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 344.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 304.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 459.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 299.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 38.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 650.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 629.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 373.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 943.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 758.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.06e3T + 9.12e5T^{2} \) |
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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
−9.520830390843841005664709362074, −8.658342086071088638391575947714, −7.67041082197706061107959641595, −6.52188093399374138284119357368, −6.08053062065311309516334960083, −5.16798785445442245162814405305, −4.04002870867959178482522444815, −3.68572564485449781159677582340, −2.47273546560750848917608531642, −0.44512995200259987297710607289,
0.44512995200259987297710607289, 2.47273546560750848917608531642, 3.68572564485449781159677582340, 4.04002870867959178482522444815, 5.16798785445442245162814405305, 6.08053062065311309516334960083, 6.52188093399374138284119357368, 7.67041082197706061107959641595, 8.658342086071088638391575947714, 9.520830390843841005664709362074
