L(s) = 1 | + (−0.263 − 0.809i)2-s + (0.222 − 0.161i)4-s + (0.331 − 1.01i)5-s + (−0.878 − 0.638i)8-s + (0.309 + 0.951i)9-s − 0.912·10-s + (−0.425 − 0.904i)11-s + (−0.115 − 0.356i)13-s + (−0.200 + 0.617i)16-s + (0.688 − 0.500i)18-s + (−0.101 − 0.0738i)19-s + (−0.0910 − 0.280i)20-s + (−0.620 + 0.582i)22-s + 0.125·23-s + (−0.120 − 0.0872i)25-s + (−0.258 + 0.187i)26-s + ⋯ |
L(s) = 1 | + (−0.263 − 0.809i)2-s + (0.222 − 0.161i)4-s + (0.331 − 1.01i)5-s + (−0.878 − 0.638i)8-s + (0.309 + 0.951i)9-s − 0.912·10-s + (−0.425 − 0.904i)11-s + (−0.115 − 0.356i)13-s + (−0.200 + 0.617i)16-s + (0.688 − 0.500i)18-s + (−0.101 − 0.0738i)19-s + (−0.0910 − 0.280i)20-s + (−0.620 + 0.582i)22-s + 0.125·23-s + (−0.120 − 0.0872i)25-s + (−0.258 + 0.187i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9608508398\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9608508398\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.425 + 0.904i)T \) |
| 79 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (0.263 + 0.809i)T + (-0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.331 + 1.01i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.115 + 0.356i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.101 + 0.0738i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - 0.125T + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.450 - 1.38i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + 1.85T + T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-1.03 + 0.749i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 - 1.75T + T^{2} \) |
| 97 | \( 1 + (-0.598 - 1.84i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−10.40099236174471925028640486213, −9.285987206819581927536750986292, −8.637095575351563692571519408228, −7.74473111739701357645026837829, −6.55540913070250209180522833895, −5.50818997928742753321105372463, −4.84325020704586923065868408539, −3.37793155177891328252425937204, −2.27147826417288976803704682619, −1.11424292704568100250996511998,
2.19890560075597474813551305929, 3.16464825335285970316204618076, 4.44760168909230943961778375461, 5.85132434324563329891801850550, 6.50061514338033425127027462578, 7.17266096644870657146129865968, 7.80437604820464926458442542550, 8.961177836533504389009987410449, 9.709017833124843879119395038556, 10.49808222952940201131839827685