L(s) = 1 | − 2.58i·2-s − i·3-s − 4.68·4-s + 1.50i·5-s − 2.58·6-s + 6.95i·8-s − 9-s + 3.88·10-s + (−0.276 + 3.30i)11-s + 4.68i·12-s + 3.99·13-s + 1.50·15-s + 8.61·16-s + 2.76·17-s + 2.58i·18-s − 4.91·19-s + ⋯ |
L(s) = 1 | − 1.82i·2-s − 0.577i·3-s − 2.34·4-s + 0.671i·5-s − 1.05·6-s + 2.45i·8-s − 0.333·9-s + 1.22·10-s + (−0.0832 + 0.996i)11-s + 1.35i·12-s + 1.10·13-s + 0.387·15-s + 2.15·16-s + 0.670·17-s + 0.609i·18-s − 1.12·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.421289373\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.421289373\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (0.276 - 3.30i)T \) |
good | 2 | \( 1 + 2.58iT - 2T^{2} \) |
| 5 | \( 1 - 1.50iT - 5T^{2} \) |
| 13 | \( 1 - 3.99T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 + 4.91T + 19T^{2} \) |
| 23 | \( 1 + 0.0719T + 23T^{2} \) |
| 29 | \( 1 + 5.06iT - 29T^{2} \) |
| 31 | \( 1 + 10.2iT - 31T^{2} \) |
| 37 | \( 1 - 7.81T + 37T^{2} \) |
| 41 | \( 1 - 2.98T + 41T^{2} \) |
| 43 | \( 1 - 5.41iT - 43T^{2} \) |
| 47 | \( 1 + 5.18iT - 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 6.37iT - 59T^{2} \) |
| 61 | \( 1 - 1.37T + 61T^{2} \) |
| 67 | \( 1 + 4.55T + 67T^{2} \) |
| 71 | \( 1 - 9.59T + 71T^{2} \) |
| 73 | \( 1 + 8.96T + 73T^{2} \) |
| 79 | \( 1 - 11.6iT - 79T^{2} \) |
| 83 | \( 1 - 9.34T + 83T^{2} \) |
| 89 | \( 1 - 13.6iT - 89T^{2} \) |
| 97 | \( 1 + 2.63iT - 97T^{2} \) |
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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
−9.434136801680634846909167686675, −8.446798430894505603881055996997, −7.75395308244479956928375427108, −6.61274926433478880480036437697, −5.71794878888796687332181486177, −4.42770060274083456445917373070, −3.77991296634822345854476502802, −2.65101748068353900684254334533, −2.04832467268825317394797861906, −0.809909858392383393947477168334,
0.911318563422256463006332364152, 3.29943407157974089492674870220, 4.21443626336360271163756488157, 5.01789981488080069247730872200, 5.76066161811648586642826336352, 6.32537055142814398117477924096, 7.29569333573543173914247518573, 8.300405168196347515568741319704, 8.730618252704587602848496658704, 9.098534944716292298341854704341