| L(s) = 1 | + 0.867i·3-s + 0.687i·5-s + (2.28 − 1.33i)7-s + 2.24·9-s + (3.31 + 0.0124i)11-s + 13-s − 0.596·15-s − 3.72·17-s − 1.66·19-s + (1.15 + 1.98i)21-s + 4.80·23-s + 4.52·25-s + 4.55i·27-s + 10.1i·29-s − 10.4i·31-s + ⋯ |
| L(s) = 1 | + 0.500i·3-s + 0.307i·5-s + (0.864 − 0.503i)7-s + 0.749·9-s + (0.999 + 0.00375i)11-s + 0.277·13-s − 0.154·15-s − 0.903·17-s − 0.381·19-s + (0.251 + 0.432i)21-s + 1.00·23-s + 0.905·25-s + 0.875i·27-s + 1.88i·29-s − 1.87i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.585878550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.585878550\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.28 + 1.33i)T \) |
| 11 | \( 1 + (-3.31 - 0.0124i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.867iT - 3T^{2} \) |
| 5 | \( 1 - 0.687iT - 5T^{2} \) |
| 17 | \( 1 + 3.72T + 17T^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 23 | \( 1 - 4.80T + 23T^{2} \) |
| 29 | \( 1 - 10.1iT - 29T^{2} \) |
| 31 | \( 1 + 10.4iT - 31T^{2} \) |
| 37 | \( 1 + 2.15T + 37T^{2} \) |
| 41 | \( 1 - 5.48T + 41T^{2} \) |
| 43 | \( 1 - 0.528iT - 43T^{2} \) |
| 47 | \( 1 + 1.89iT - 47T^{2} \) |
| 53 | \( 1 + 4.92T + 53T^{2} \) |
| 59 | \( 1 - 3.88iT - 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 2.15T + 71T^{2} \) |
| 73 | \( 1 - 1.25T + 73T^{2} \) |
| 79 | \( 1 + 6.76iT - 79T^{2} \) |
| 83 | \( 1 + 5.89T + 83T^{2} \) |
| 89 | \( 1 + 12.7iT - 89T^{2} \) |
| 97 | \( 1 + 7.61iT - 97T^{2} \) |
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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
−8.728970018044528464981044915015, −7.68096736635954911778305226129, −6.98449566748900309002725544665, −6.52704202168539123277213401667, −5.37035399106365042936062110863, −4.54037878630830292966769719691, −4.11351854040759608459502193570, −3.20238849475859411259195264008, −1.93211106813413136934516799382, −1.03375848278896147187253172648,
0.987544350627231890173416321277, 1.72359214647610998029177274662, 2.67166779327965665625977350261, 3.97530606055918986059521744123, 4.59460191577076467910072212188, 5.32480745251191479259805881380, 6.46118647708743944004257042057, 6.75799217084249771597764996681, 7.70287929678134315615363878220, 8.436364950394860213063695993344
