| L(s) = 1 | − 3·3-s − 14.2·5-s + 1.25·7-s + 9·9-s − 7.74·13-s + 42.7·15-s + 99.8·17-s + 34.2·19-s − 3.77·21-s − 79.5·23-s + 78.2·25-s − 27·27-s − 234.·29-s − 201.·31-s − 17.9·35-s − 358.·37-s + 23.2·39-s + 20.3·41-s − 73.1·43-s − 128.·45-s + 121.·47-s − 341.·49-s − 299.·51-s − 63.7·53-s − 102.·57-s − 10.2·59-s + 713.·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.27·5-s + 0.0679·7-s + 0.333·9-s − 0.165·13-s + 0.736·15-s + 1.42·17-s + 0.414·19-s − 0.0392·21-s − 0.721·23-s + 0.626·25-s − 0.192·27-s − 1.50·29-s − 1.16·31-s − 0.0866·35-s − 1.59·37-s + 0.0953·39-s + 0.0776·41-s − 0.259·43-s − 0.425·45-s + 0.377·47-s − 0.995·49-s − 0.822·51-s − 0.165·53-s − 0.239·57-s − 0.0226·59-s + 1.49·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8149718620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8149718620\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 14.2T + 125T^{2} \) |
| 7 | \( 1 - 1.25T + 343T^{2} \) |
| 13 | \( 1 + 7.74T + 2.19e3T^{2} \) |
| 17 | \( 1 - 99.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 79.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 234.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 201.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 358.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 20.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 73.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 121.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 63.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 10.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 713.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 288.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 149.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 67.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 935.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 595.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 136.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.37e3T + 9.12e5T^{2} \) |
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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
−9.192439566385050322436710354872, −8.086726293364734876071453100691, −7.59876356364083105868105310981, −6.88123267104503456299721567783, −5.69427976537172253736250011912, −5.08016544940087750173252316834, −3.89494708908642172912641381601, −3.40529770727421548419385340634, −1.76326165029079916842098763825, −0.45749852288396485706045179422,
0.45749852288396485706045179422, 1.76326165029079916842098763825, 3.40529770727421548419385340634, 3.89494708908642172912641381601, 5.08016544940087750173252316834, 5.69427976537172253736250011912, 6.88123267104503456299721567783, 7.59876356364083105868105310981, 8.086726293364734876071453100691, 9.192439566385050322436710354872
