| L(s) = 1 | − i·2-s + 2i·3-s − 4-s + 2·6-s − i·7-s + i·8-s − 9-s + 11-s − 2i·12-s − 14-s + 16-s + i·18-s + 2·21-s − i·22-s + 4i·23-s − 2·24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.15i·3-s − 0.5·4-s + 0.816·6-s − 0.377i·7-s + 0.353i·8-s − 0.333·9-s + 0.301·11-s − 0.577i·12-s − 0.267·14-s + 0.250·16-s + 0.235i·18-s + 0.436·21-s − 0.213i·22-s + 0.834i·23-s − 0.408·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.587757549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.587757549\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.102851016375685355449668421369, −7.965457524100276422249051363171, −7.31002701081667890797758403167, −6.22503240157641070425265637778, −5.34439405294479514502170961882, −4.64201960507548573648465988986, −3.88833155141469657077173692272, −3.42073639932996228730471607585, −2.26480245974973972698210904445, −1.06917646227446723916463943455,
0.53891349391005806364379110360, 1.69134497070612708169180493123, 2.62971618310070572755495447571, 3.81766246686068104111632138353, 4.71318187376635366427411527391, 5.65592281802215143797257099778, 6.27522855776829928288571794061, 6.94414321189113682566220335605, 7.46968820428559218939756119381, 8.249730638991834068677682197615
