| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + i·7-s − i·8-s − 9-s + 2·11-s + i·12-s + i·13-s − 14-s + 16-s + 3i·17-s − i·18-s + 21-s + 2i·22-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.377i·7-s − 0.353i·8-s − 0.333·9-s + 0.603·11-s + 0.288i·12-s + 0.277i·13-s − 0.267·14-s + 0.250·16-s + 0.727i·17-s − 0.235i·18-s + 0.218·21-s + 0.426i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.490472834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.490472834\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 - 11iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 - 5T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 11iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.808632460466585897765817262420, −9.073971963379388790920887533642, −8.247290282532576061032369182949, −7.61818271994208060256571797231, −6.45844813884723478152021312953, −6.19468258290892207301574983040, −4.98767384158001621680400013200, −4.01042146023349905158645309673, −2.68583287761610069750233944860, −1.22181019720125602154258706613,
0.811903145087288829486097771831, 2.43335352298461453416479505235, 3.48072915361499806176663674517, 4.36236858430158973689928417783, 5.17112473360551779962615920756, 6.30124024137353588011981718323, 7.31652005858638648870522305525, 8.420874344238838667094674180146, 9.055382572194412748832084444824, 10.04203206286010946799635660096
