| L(s) = 1 | + (−1.41 + i)3-s − 5-s + 2i·7-s + (1.00 − 2.82i)9-s − 2i·11-s − 2.82i·13-s + (1.41 − i)15-s − 2.82i·17-s − 5.65·19-s + (−2 − 2.82i)21-s + 8.48·23-s + 25-s + (1.41 + 5.00i)27-s + 2·29-s + 6i·31-s + ⋯ |
| L(s) = 1 | + (−0.816 + 0.577i)3-s − 0.447·5-s + 0.755i·7-s + (0.333 − 0.942i)9-s − 0.603i·11-s − 0.784i·13-s + (0.365 − 0.258i)15-s − 0.685i·17-s − 1.29·19-s + (−0.436 − 0.617i)21-s + 1.76·23-s + 0.200·25-s + (0.272 + 0.962i)27-s + 0.371·29-s + 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.979150 - 0.0833885i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.979150 - 0.0833885i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 - i)T \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 - 2.82iT - 37T^{2} \) |
| 41 | \( 1 + 11.3iT - 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 10iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 2.82T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−10.23898573331409105139420014192, −8.971205067848928209267904614535, −8.693618753156257392459189515903, −7.34507769157533859461863397456, −6.48390110024736871843362987081, −5.52085984458008069685902947328, −4.91146767554922102664139625577, −3.75427453299980610537228553763, −2.71922365485351590949553886827, −0.67142849882816230004825533660,
1.00505730797205185834451090327, 2.35750430592623377881582532157, 4.09208062943069456411733875549, 4.62127726278236608139524778222, 5.88123190369505150680465961096, 6.79581718641471765254153249095, 7.29584987942659229016975095373, 8.207759066672277694012149034381, 9.212163792102632676134198593407, 10.34399674723421607196955900551
