| L(s) = 1 | − 5.51i·3-s − 78.3·7-s + 50.5·9-s − 123.·11-s − 229.·13-s − 557. i·17-s − 334.·19-s + 432. i·21-s + 244.·23-s − 725. i·27-s − 301. i·29-s + 339. i·31-s + 679. i·33-s − 187.·37-s + 1.26e3i·39-s + ⋯ |
| L(s) = 1 | − 0.612i·3-s − 1.59·7-s + 0.624·9-s − 1.01·11-s − 1.35·13-s − 1.92i·17-s − 0.925·19-s + 0.979i·21-s + 0.461·23-s − 0.995i·27-s − 0.358i·29-s + 0.353i·31-s + 0.623i·33-s − 0.136·37-s + 0.833i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5203742043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5203742043\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5.51iT - 81T^{2} \) |
| 7 | \( 1 + 78.3T + 2.40e3T^{2} \) |
| 11 | \( 1 + 123.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 229.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 557. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 334.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 244.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 301. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 339. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 187.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.28e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.07e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 3.24e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 2.02e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 555.T + 1.21e7T^{2} \) |
| 61 | \( 1 - 3.18e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 3.43e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.40e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.09e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 2.93e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 8.32e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 406.T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.76e4iT - 8.85e7T^{2} \) |
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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
−9.836417574925723879049887923168, −9.123474446516929864670318170169, −7.80720187751612523385196696739, −7.10362638612391563630577190678, −6.59286934115549429079234114463, −5.38312961163151162114270689754, −4.43167616778088620790112964280, −2.96796671133221929814344285677, −2.38291132901383866504246349141, −0.62927297853766497970783662555,
0.18338564345894239837767431180, 2.05776900879149324239211520270, 3.18891279050029484793951743895, 4.07366436288220159968754680414, 5.05895267911823616560084834703, 6.13622978968147202680424011778, 6.92351106188002632007933115995, 7.87368492802795559430025558445, 8.951744439821716253880817342554, 9.821587258753542143956125951188
