| L(s) = 1 | + 2·3-s + i·5-s + i·7-s + 9-s − i·11-s + (2 + 3i)13-s + 2i·15-s − 4·17-s + 5i·19-s + 2i·21-s − 9·23-s + 4·25-s − 4·27-s + 5·29-s + 7i·31-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447i·5-s + 0.377i·7-s + 0.333·9-s − 0.301i·11-s + (0.554 + 0.832i)13-s + 0.516i·15-s − 0.970·17-s + 1.14i·19-s + 0.436i·21-s − 1.87·23-s + 0.800·25-s − 0.769·27-s + 0.928·29-s + 1.25i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.896918708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.896918708\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (-2 - 3i)T \) |
| good | 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 - iT - 5T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 5iT - 19T^{2} \) |
| 23 | \( 1 + 9T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 7iT - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 + iT - 47T^{2} \) |
| 53 | \( 1 + T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 4iT - 71T^{2} \) |
| 73 | \( 1 - 3iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 15iT - 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−8.575743148011344027513333318159, −8.276776123892529987888694711226, −7.32178642999868261558876136403, −6.49192735877984206602136141247, −5.96060126564539144458662538490, −4.80394909629311629806279478012, −3.86022725976455343925621122954, −3.28119953419182307921445400578, −2.34877534636782024729746803760, −1.63987801181043504309223271180,
0.41808378079955342218917946664, 1.80918778310023956417071817803, 2.65203919939696281157628525320, 3.47123747291992944701967450179, 4.31880604417427468245002468009, 4.99163843444449487612549555986, 6.11909421275778372728999523173, 6.75855509489159911853117988948, 7.87681678892196189220184619832, 8.099691654171522620558566720022
