| L(s) = 1 | − 0.950i·3-s + 3.52i·5-s + (2.60 − 0.469i)7-s + 2.09·9-s + (−2.46 + 2.21i)11-s − 13-s + 3.35·15-s + 2.15·17-s + 4.00·19-s + (−0.446 − 2.47i)21-s + 3.36·23-s − 7.45·25-s − 4.84i·27-s + 7.21i·29-s − 6.56i·31-s + ⋯ |
| L(s) = 1 | − 0.548i·3-s + 1.57i·5-s + (0.984 − 0.177i)7-s + 0.699·9-s + (−0.744 + 0.667i)11-s − 0.277·13-s + 0.866·15-s + 0.523·17-s + 0.919·19-s + (−0.0973 − 0.539i)21-s + 0.700·23-s − 1.49·25-s − 0.932i·27-s + 1.33i·29-s − 1.17i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.334893522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.334893522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.60 + 0.469i)T \) |
| 11 | \( 1 + (2.46 - 2.21i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.950iT - 3T^{2} \) |
| 5 | \( 1 - 3.52iT - 5T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 19 | \( 1 - 4.00T + 19T^{2} \) |
| 23 | \( 1 - 3.36T + 23T^{2} \) |
| 29 | \( 1 - 7.21iT - 29T^{2} \) |
| 31 | \( 1 + 6.56iT - 31T^{2} \) |
| 37 | \( 1 - 9.55T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 + 1.28iT - 43T^{2} \) |
| 47 | \( 1 - 4.74iT - 47T^{2} \) |
| 53 | \( 1 + 6.72T + 53T^{2} \) |
| 59 | \( 1 + 3.55iT - 59T^{2} \) |
| 61 | \( 1 - 9.88T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 5.39T + 71T^{2} \) |
| 73 | \( 1 + 4.53T + 73T^{2} \) |
| 79 | \( 1 - 0.287iT - 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 - 9.11iT - 89T^{2} \) |
| 97 | \( 1 - 10.6iT - 97T^{2} \) |
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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
−8.120638795355601682439250409230, −7.68275642144264859051186528100, −7.14608878738292551788371505493, −6.67210256911999784483246864461, −5.60131832292191348172915400977, −4.85174843166364756939552592479, −3.90707018280540838391101996502, −2.88195656905714235833077054634, −2.20221325952406483147308734675, −1.16422122690267038669110507983,
0.78787805987142123591357454415, 1.56794626979616113602797467448, 2.85217370731988108712229118337, 4.01018773056950286173358234695, 4.68185649812410958787808112511, 5.24181639932960319220964938534, 5.68296271523157916857988184179, 7.04050729147155772981576520229, 7.943005780781855711100027497155, 8.256720053731148049306584434499
