| L(s) = 1 | − 1.41i·2-s + 2.23i·5-s + (−1.58 − 2.12i)7-s − 2.82i·8-s + 3.16·10-s + (3 + 1.41i)11-s + 6.32·13-s + (−3 + 2.23i)14-s − 4.00·16-s − 3.16·19-s + (2.00 − 4.24i)22-s + 3·23-s − 8.94i·26-s − 1.41i·29-s − 6.70i·31-s + ⋯ |
| L(s) = 1 | − 0.999i·2-s + 0.999i·5-s + (−0.597 − 0.801i)7-s − 0.999i·8-s + 1.00·10-s + (0.904 + 0.426i)11-s + 1.75·13-s + (−0.801 + 0.597i)14-s − 1.00·16-s − 0.725·19-s + (0.426 − 0.904i)22-s + 0.625·23-s − 1.75i·26-s − 0.262i·29-s − 1.20i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35289 - 1.10615i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35289 - 1.10615i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.58 + 2.12i)T \) |
| 11 | \( 1 + (-3 - 1.41i)T \) |
| good | 2 | \( 1 + 1.41iT - 2T^{2} \) |
| 5 | \( 1 - 2.23iT - 5T^{2} \) |
| 13 | \( 1 - 6.32T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 6.70iT - 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 9.48T + 41T^{2} \) |
| 43 | \( 1 + 4.24iT - 43T^{2} \) |
| 47 | \( 1 + 4.47iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 2.23iT - 59T^{2} \) |
| 61 | \( 1 + 3.16T + 61T^{2} \) |
| 67 | \( 1 - 11T + 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + 3.16T + 73T^{2} \) |
| 79 | \( 1 - 8.48iT - 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 - 2.23iT - 89T^{2} \) |
| 97 | \( 1 - 6.70iT - 97T^{2} \) |
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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
−10.55124139294434507417266434912, −9.724682006743377061000053877322, −8.850727607459670890698225659316, −7.45200242418742605998078208163, −6.64643807124698872723829690981, −6.16425291141987823577459647405, −4.08006956632318867808573560238, −3.64118611163535897157739796464, −2.51090540957643382877740457688, −1.11596597512819646514742270536,
1.41551866160444577562469600307, 3.10748711872335590498202063119, 4.43310233732232086667535734758, 5.59878409558956006049679718773, 6.17646544860301489687231350144, 6.90573463110059722313200561241, 8.298329513742875838437962411125, 8.727708633457307222816468131450, 9.273213059061316939275778171237, 10.79802605461915406021708383686
