L(s) = 1 | + 2.75i·2-s − 5.61·4-s + 2.24·5-s − 9.97i·8-s + 6.19i·10-s + 0.797i·11-s + 5.06i·13-s + 16.2·16-s − 4.46·17-s − 1.25i·19-s − 12.6·20-s − 2.20·22-s + 3.78i·23-s + 0.0366·25-s − 13.9·26-s + ⋯ |
L(s) = 1 | + 1.95i·2-s − 2.80·4-s + 1.00·5-s − 3.52i·8-s + 1.95i·10-s + 0.240i·11-s + 1.40i·13-s + 4.07·16-s − 1.08·17-s − 0.287i·19-s − 2.81·20-s − 0.469·22-s + 0.789i·23-s + 0.00733·25-s − 2.73·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.013153786\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.013153786\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.75iT - 2T^{2} \) |
| 5 | \( 1 - 2.24T + 5T^{2} \) |
| 11 | \( 1 - 0.797iT - 11T^{2} \) |
| 13 | \( 1 - 5.06iT - 13T^{2} \) |
| 17 | \( 1 + 4.46T + 17T^{2} \) |
| 19 | \( 1 + 1.25iT - 19T^{2} \) |
| 23 | \( 1 - 3.78iT - 23T^{2} \) |
| 29 | \( 1 - 4.04iT - 29T^{2} \) |
| 31 | \( 1 - 5.35iT - 31T^{2} \) |
| 37 | \( 1 + 5.33T + 37T^{2} \) |
| 41 | \( 1 + 4.50T + 41T^{2} \) |
| 43 | \( 1 - 0.390T + 43T^{2} \) |
| 47 | \( 1 + 6.60T + 47T^{2} \) |
| 53 | \( 1 + 6.67iT - 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 8.12iT - 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 3.19iT - 71T^{2} \) |
| 73 | \( 1 + 9.03iT - 73T^{2} \) |
| 79 | \( 1 - 5.14T + 79T^{2} \) |
| 83 | \( 1 - 9.29T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 1.93iT - 97T^{2} \) |
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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
−9.645566969957592555037654643653, −9.146483561126256901803495368486, −8.589904317614626043531134742676, −7.50652795438602766802808743686, −6.66221094229418836482012378517, −6.41119593450548435333466732071, −5.25966867179630434578325266036, −4.75896578038184038155026595263, −3.65798952221618000978078440522, −1.73988765454096798368312236871,
0.41738980518762755904305415103, 1.80337820866212289761595151019, 2.57951280097128300443708685929, 3.50316475234936799584791368206, 4.57097589631195192053829371751, 5.41050450897024456382337795055, 6.24960269065203903985699299388, 7.950492450531561607633961448246, 8.601719507679066556542881523992, 9.512904631443609880878438125612