| L(s) = 1 | − 1.76i·3-s − 1.09i·5-s + (−1.82 + 1.91i)7-s − 0.106·9-s + (3.28 − 0.475i)11-s + 13-s − 1.93·15-s + 3.56·17-s − 6.44·19-s + (3.38 + 3.21i)21-s − 7.65·23-s + 3.79·25-s − 5.10i·27-s + 4.80i·29-s − 2.61i·31-s + ⋯ |
| L(s) = 1 | − 1.01i·3-s − 0.490i·5-s + (−0.688 + 0.725i)7-s − 0.0354·9-s + (0.989 − 0.143i)11-s + 0.277·13-s − 0.499·15-s + 0.863·17-s − 1.47·19-s + (0.737 + 0.700i)21-s − 1.59·23-s + 0.759·25-s − 0.981i·27-s + 0.892i·29-s − 0.469i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.578731994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.578731994\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.82 - 1.91i)T \) |
| 11 | \( 1 + (-3.28 + 0.475i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.76iT - 3T^{2} \) |
| 5 | \( 1 + 1.09iT - 5T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 - 4.80iT - 29T^{2} \) |
| 31 | \( 1 + 2.61iT - 31T^{2} \) |
| 37 | \( 1 + 0.983T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 9.59iT - 43T^{2} \) |
| 47 | \( 1 + 0.484iT - 47T^{2} \) |
| 53 | \( 1 - 6.12T + 53T^{2} \) |
| 59 | \( 1 + 12.7iT - 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 6.00T + 67T^{2} \) |
| 71 | \( 1 - 7.48T + 71T^{2} \) |
| 73 | \( 1 - 3.59T + 73T^{2} \) |
| 79 | \( 1 - 0.469iT - 79T^{2} \) |
| 83 | \( 1 + 0.346T + 83T^{2} \) |
| 89 | \( 1 + 13.1iT - 89T^{2} \) |
| 97 | \( 1 + 15.8iT - 97T^{2} \) |
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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
−8.266436508798688583387471215840, −7.43073500550307915790847345879, −6.56997646014026004953872232781, −6.20467954476224634863375816458, −5.44351841132655549695882830534, −4.27789810985955438763086330394, −3.57234205030564279042212498422, −2.35618277764239872254601387244, −1.62272571419736087874585422350, −0.50102496924042858750020741987,
1.14938374839850481647286869851, 2.52491887977198732033150430231, 3.60475570307365883361464969356, 4.04084323331856888069080540389, 4.62904217886203666652714664629, 5.94127262130667172723212314697, 6.38654004964303936529498137962, 7.17866965318552618210521143747, 7.972258993494528262297875559310, 8.874732054340598810811766565216
