| L(s) = 1 | − i·2-s + (−1.47 − 0.907i)3-s − 4-s − 5-s + (−0.907 + 1.47i)6-s + i·8-s + (1.35 + 2.67i)9-s + i·10-s − 3.28i·11-s + (1.47 + 0.907i)12-s − 5.91i·13-s + (1.47 + 0.907i)15-s + 16-s − 2.40·17-s + (2.67 − 1.35i)18-s − 5.51i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.851 − 0.523i)3-s − 0.5·4-s − 0.447·5-s + (−0.370 + 0.602i)6-s + 0.353i·8-s + (0.451 + 0.892i)9-s + 0.316i·10-s − 0.990i·11-s + (0.425 + 0.261i)12-s − 1.64i·13-s + (0.380 + 0.234i)15-s + 0.250·16-s − 0.583·17-s + (0.631 − 0.319i)18-s − 1.26i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2294562675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2294562675\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (1.47 + 0.907i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3.28iT - 11T^{2} \) |
| 13 | \( 1 + 5.91iT - 13T^{2} \) |
| 17 | \( 1 + 2.40T + 17T^{2} \) |
| 19 | \( 1 + 5.51iT - 19T^{2} \) |
| 23 | \( 1 - 6.49iT - 23T^{2} \) |
| 29 | \( 1 + 3.80iT - 29T^{2} \) |
| 31 | \( 1 - 5.20iT - 31T^{2} \) |
| 37 | \( 1 + 1.88T + 37T^{2} \) |
| 41 | \( 1 + 0.103T + 41T^{2} \) |
| 43 | \( 1 + 1.48T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 2.43iT - 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 - 14.2iT - 61T^{2} \) |
| 67 | \( 1 + 3.35T + 67T^{2} \) |
| 71 | \( 1 - 13.9iT - 71T^{2} \) |
| 73 | \( 1 + 8.11iT - 73T^{2} \) |
| 79 | \( 1 - 9.15T + 79T^{2} \) |
| 83 | \( 1 + 6.54T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 7.90iT - 97T^{2} \) |
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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
−8.890900039094374478476237780195, −8.117922147617507281855863590713, −7.39420839698097010315859416509, −6.38278973901083005352464315571, −5.46422637474978594130064147333, −4.84413647294385757316058304934, −3.57093252768116768208449986938, −2.68145802822024154430827233329, −1.17645059182971112430201382390, −0.11617200287477059022699813256,
1.79327253672249805856801618601, 3.66821467715584431029106636303, 4.47814398805493643372319693219, 4.92722184135318751274238869247, 6.25173684504775060275394279479, 6.63855674537658249761414361415, 7.49290872115269229593052865338, 8.499685719656012796343496293116, 9.341109356160291648880533321369, 9.938515199151744803149250263945
