| L(s) = 1 | + 2.73i·3-s + 3.14i·5-s + (1.11 − 2.39i)7-s − 4.49·9-s + (2.27 + 2.40i)11-s − 13-s − 8.61·15-s + 3.90·17-s + 1.86·19-s + (6.56 + 3.04i)21-s − 1.60·23-s − 4.90·25-s − 4.08i·27-s + 6.06i·29-s + 4.12i·31-s + ⋯ |
| L(s) = 1 | + 1.58i·3-s + 1.40i·5-s + (0.420 − 0.907i)7-s − 1.49·9-s + (0.687 + 0.726i)11-s − 0.277·13-s − 2.22·15-s + 0.945·17-s + 0.427·19-s + (1.43 + 0.665i)21-s − 0.333·23-s − 0.981·25-s − 0.785i·27-s + 1.12i·29-s + 0.741i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.720400099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.720400099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.11 + 2.39i)T \) |
| 11 | \( 1 + (-2.27 - 2.40i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.73iT - 3T^{2} \) |
| 5 | \( 1 - 3.14iT - 5T^{2} \) |
| 17 | \( 1 - 3.90T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 + 1.60T + 23T^{2} \) |
| 29 | \( 1 - 6.06iT - 29T^{2} \) |
| 31 | \( 1 - 4.12iT - 31T^{2} \) |
| 37 | \( 1 - 3.63T + 37T^{2} \) |
| 41 | \( 1 + 3.75T + 41T^{2} \) |
| 43 | \( 1 - 11.3iT - 43T^{2} \) |
| 47 | \( 1 + 11.1iT - 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 10.2iT - 59T^{2} \) |
| 61 | \( 1 - 4.42T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 4.22T + 71T^{2} \) |
| 73 | \( 1 - 0.120T + 73T^{2} \) |
| 79 | \( 1 - 5.64iT - 79T^{2} \) |
| 83 | \( 1 + 5.02T + 83T^{2} \) |
| 89 | \( 1 + 14.7iT - 89T^{2} \) |
| 97 | \( 1 - 16.7iT - 97T^{2} \) |
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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
−9.103842289090931850986641609471, −8.088075198505792979992100136779, −7.28324161844393466754285169461, −6.78746266778352856890104321087, −5.79631358957000747845625342305, −4.86255264384835774187640533856, −4.30321114070213066769548939517, −3.41775872092368723306701547205, −3.03414246137372515057606434746, −1.54452223706502539071425641866,
0.51244675847091181348256882611, 1.35347499710051997531413618357, 2.06216116985882582512988728554, 3.14179284458480134718677977981, 4.37400639574652701157053427354, 5.28691173577083905617162593746, 5.90898222469583124749411501685, 6.40966529423483097415777218824, 7.68466442391006682185999638447, 7.88352135384538012773256041577
