L(s) = 1 | − 2i·3-s − i·7-s − 9-s − 11-s + 4i·13-s + 6·19-s − 2·21-s − 3i·23-s − 4i·27-s + 3·29-s + 2i·33-s − 9i·37-s + 8·39-s + 2·41-s − 9i·43-s + ⋯ |
L(s) = 1 | − 1.15i·3-s − 0.377i·7-s − 0.333·9-s − 0.301·11-s + 1.10i·13-s + 1.37·19-s − 0.436·21-s − 0.625i·23-s − 0.769i·27-s + 0.557·29-s + 0.348i·33-s − 1.47i·37-s + 1.28·39-s + 0.312·41-s − 1.37i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.719246802\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.719246802\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 9iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + iT - 67T^{2} \) |
| 71 | \( 1 - 7T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 9T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−8.470406925835257224876919855044, −7.49055053920893574422695295191, −7.22005348022164819018215937340, −6.45339982788102872040577029105, −5.62593631804495930900112961848, −4.63161366664251013192301746204, −3.73385080153482245591792724714, −2.55084998666211776549106254303, −1.68519892909677200909587451682, −0.61781531551502741500821885337,
1.20961378144737726799388679502, 2.83559990731355648314662776524, 3.34485876795042671043189042136, 4.41292893251805478722163867192, 5.17070944203388447469133486069, 5.66599678739973971892782164914, 6.73457926145965566675673375587, 7.75147914060260894247591903181, 8.271173931353548175343207466521, 9.305269082379328462856966552538