| L(s) = 1 | − 2.95i·3-s + 1.47i·5-s + (0.700 + 2.55i)7-s − 5.71·9-s + (−2.46 + 2.21i)11-s + 13-s + 4.35·15-s − 0.856·17-s + 2.02·19-s + (7.53 − 2.06i)21-s − 5.44·23-s + 2.82·25-s + 8.02i·27-s − 7.20i·29-s − 8.75i·31-s + ⋯ |
| L(s) = 1 | − 1.70i·3-s + 0.659i·5-s + (0.264 + 0.964i)7-s − 1.90·9-s + (−0.743 + 0.668i)11-s + 0.277·13-s + 1.12·15-s − 0.207·17-s + 0.464·19-s + (1.64 − 0.451i)21-s − 1.13·23-s + 0.565·25-s + 1.54i·27-s − 1.33i·29-s − 1.57i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1863885201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1863885201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.700 - 2.55i)T \) |
| 11 | \( 1 + (2.46 - 2.21i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.95iT - 3T^{2} \) |
| 5 | \( 1 - 1.47iT - 5T^{2} \) |
| 17 | \( 1 + 0.856T + 17T^{2} \) |
| 19 | \( 1 - 2.02T + 19T^{2} \) |
| 23 | \( 1 + 5.44T + 23T^{2} \) |
| 29 | \( 1 + 7.20iT - 29T^{2} \) |
| 31 | \( 1 + 8.75iT - 31T^{2} \) |
| 37 | \( 1 + 0.648T + 37T^{2} \) |
| 41 | \( 1 - 0.228T + 41T^{2} \) |
| 43 | \( 1 + 3.56iT - 43T^{2} \) |
| 47 | \( 1 - 0.0660iT - 47T^{2} \) |
| 53 | \( 1 + 8.15T + 53T^{2} \) |
| 59 | \( 1 - 8.61iT - 59T^{2} \) |
| 61 | \( 1 + 2.74T + 61T^{2} \) |
| 67 | \( 1 - 2.46T + 67T^{2} \) |
| 71 | \( 1 + 2.94T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 11.5iT - 79T^{2} \) |
| 83 | \( 1 + 3.70T + 83T^{2} \) |
| 89 | \( 1 - 1.53iT - 89T^{2} \) |
| 97 | \( 1 + 1.74iT - 97T^{2} \) |
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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
−7.86691557774438749507437145125, −7.39533571397201688925460079639, −6.54840332499995073857004131567, −5.99193952319317904163031899857, −5.36846446132590440050245569093, −4.16440775648418022914957599193, −2.74929963122405871874401591822, −2.40555596829008171535637959783, −1.53161603386777259199296569000, −0.05210399423045841380590056692,
1.36609670102612511930994434864, 3.03890411243313875844264731463, 3.56057163454837137959513811020, 4.48043169498631867924194970315, 4.95257991546567715033915457260, 5.56550546733707180692887052694, 6.57038328084758300429033746540, 7.63428209153038513134675905962, 8.421745324817537186125406778904, 8.855597228087247301848921285583
