L(s) = 1 | + (−0.866 − 0.5i)3-s + (−1.86 − 1.23i)5-s + (0.499 + 0.866i)9-s + (−2 + 3.46i)11-s − 2i·13-s + (1 + 2i)15-s + (−1.73 − i)17-s + (−1 − 1.73i)19-s + (5.19 − 3i)23-s + (1.96 + 4.59i)25-s − 0.999i·27-s − 6·29-s + (−3 + 5.19i)31-s + (3.46 − 1.99i)33-s + (−3.46 + 2i)37-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.288i)3-s + (−0.834 − 0.550i)5-s + (0.166 + 0.288i)9-s + (−0.603 + 1.04i)11-s − 0.554i·13-s + (0.258 + 0.516i)15-s + (−0.420 − 0.242i)17-s + (−0.229 − 0.397i)19-s + (1.08 − 0.625i)23-s + (0.392 + 0.919i)25-s − 0.192i·27-s − 1.11·29-s + (−0.538 + 0.933i)31-s + (0.603 − 0.348i)33-s + (−0.569 + 0.328i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8881714720\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8881714720\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (1.86 + 1.23i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.46 - 2i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (-3.46 + 2i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.73 + i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.3 - 6i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (-12.1 - 7i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + (-8 - 13.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.671013120397769676899477391490, −7.945255867625545287452307043017, −7.17918783762698464022007086209, −6.76126582221572617860858173319, −5.43102910760629235656569526373, −4.98170299603507502752135541587, −4.21256462874641387481762700075, −3.12035598779514331293677031995, −1.98135810392047965131712030958, −0.68063198348365279582493861962,
0.48982844565146042410439009164, 2.12530038440409586566352640339, 3.36164025107133977274936533838, 3.86077231314207091376670457102, 4.87954980822581374168630747319, 5.70617969200202026266563422609, 6.43919723378047111977680697845, 7.28403063251357730183903230960, 7.87825002769289535079366660441, 8.785900909059721963060795621638