| L(s) = 1 | + 3.38·3-s − 0.456i·5-s − i·7-s + 8.42·9-s + i·11-s + (−3.09 − 1.85i)13-s − 1.54i·15-s − 0.655·17-s − 4.48i·19-s − 3.38i·21-s + 2.36·23-s + 4.79·25-s + 18.3·27-s + 3.97·29-s + 6.41i·31-s + ⋯ |
| L(s) = 1 | + 1.95·3-s − 0.203i·5-s − 0.377i·7-s + 2.80·9-s + 0.301i·11-s + (−0.858 − 0.513i)13-s − 0.398i·15-s − 0.159·17-s − 1.02i·19-s − 0.737i·21-s + 0.492·23-s + 0.958·25-s + 3.53·27-s + 0.738·29-s + 1.15i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.211326738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.211326738\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (3.09 + 1.85i)T \) |
| good | 3 | \( 1 - 3.38T + 3T^{2} \) |
| 5 | \( 1 + 0.456iT - 5T^{2} \) |
| 17 | \( 1 + 0.655T + 17T^{2} \) |
| 19 | \( 1 + 4.48iT - 19T^{2} \) |
| 23 | \( 1 - 2.36T + 23T^{2} \) |
| 29 | \( 1 - 3.97T + 29T^{2} \) |
| 31 | \( 1 - 6.41iT - 31T^{2} \) |
| 37 | \( 1 - 5.80iT - 37T^{2} \) |
| 41 | \( 1 + 4.28iT - 41T^{2} \) |
| 43 | \( 1 - 1.05T + 43T^{2} \) |
| 47 | \( 1 - 2.20iT - 47T^{2} \) |
| 53 | \( 1 - 2.63T + 53T^{2} \) |
| 59 | \( 1 + 4.87iT - 59T^{2} \) |
| 61 | \( 1 + 1.09T + 61T^{2} \) |
| 67 | \( 1 + 9.64iT - 67T^{2} \) |
| 71 | \( 1 + 5.47iT - 71T^{2} \) |
| 73 | \( 1 + 16.3iT - 73T^{2} \) |
| 79 | \( 1 - 9.07T + 79T^{2} \) |
| 83 | \( 1 + 0.226iT - 83T^{2} \) |
| 89 | \( 1 + 13.9iT - 89T^{2} \) |
| 97 | \( 1 - 15.7iT - 97T^{2} \) |
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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
−8.405927709212889016441068107341, −7.77976341874563881477067040498, −7.10006903040983991231981349838, −6.62064604221132674645946736362, −4.84067852770106335208982218799, −4.71040204129908382210482342358, −3.47265593102314343217818070903, −2.92428888713643252824575755779, −2.13853274926523779253703182197, −1.02152247898120484206072485515,
1.33364172285785466665295217106, 2.42392283816669464817840810727, 2.78982575551838376435698645843, 3.82642900492639068490260408199, 4.39298066596674353495130566225, 5.45729265058461551801473855389, 6.65065421823470171500205277519, 7.20098503231136077997725455601, 7.959806992910924967872216649123, 8.498221255511452225345632376683
