| L(s) = 1 | − 5.40i·2-s − 21.2·4-s − 18.5·7-s + 71.5i·8-s − 66.7i·11-s + 100. i·14-s + 216.·16-s − 361.·22-s − 125. i·23-s − 125·25-s + 393.·28-s − 69.7i·29-s − 600. i·32-s + 10.5·37-s + 534.·43-s + 1.41e3i·44-s + ⋯ |
| L(s) = 1 | − 1.91i·2-s − 2.65·4-s − 0.999·7-s + 3.16i·8-s − 1.83i·11-s + 1.91i·14-s + 3.38·16-s − 3.49·22-s − 1.13i·23-s − 25-s + 2.65·28-s − 0.446i·29-s − 3.31i·32-s + 0.0470·37-s + 1.89·43-s + 4.85i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.333519 + 0.644309i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.333519 + 0.644309i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 18.5T \) |
| good | 2 | \( 1 + 5.40iT - 8T^{2} \) |
| 5 | \( 1 + 125T^{2} \) |
| 11 | \( 1 + 66.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 125. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 69.7iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 - 10.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 534.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 65.4iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 740T + 3.00e5T^{2} \) |
| 71 | \( 1 + 205. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.38e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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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
−13.40611239178433023752164801523, −12.49424301303578441900398268378, −11.39959027761754554692468351354, −10.51009801635924384486448848235, −9.420753723152216723632914325286, −8.402392327330469460790107586973, −5.85680225559852868161634825107, −3.93566870972386500702500289583, −2.76272598405608119082452877240, −0.50827161462858869577201668080,
4.14452910051773280477311402234, 5.59328794845104549216704373775, 6.85882528085845714350427961432, 7.65469449645162492878361295795, 9.241450780954893282778454900706, 9.923477192646726845296226898316, 12.41587010489270299267074588621, 13.28830523720173095940421607502, 14.41324914033663825553491800872, 15.44607044552172187309435252730
