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
−8.785900909059721963060795621638, −7.87825002769289535079366660441, −7.28403063251357730183903230960, −6.43919723378047111977680697845, −5.70617969200202026266563422609, −4.87954980822581374168630747319, −3.86077231314207091376670457102, −3.36164025107133977274936533838, −2.12530038440409586566352640339, −0.48982844565146042410439009164,
0.68063198348365279582493861962, 1.98135810392047965131712030958, 3.12035598779514331293677031995, 4.21256462874641387481762700075, 4.98170299603507502752135541587, 5.43102910760629235656569526373, 6.76126582221572617860858173319, 7.17918783762698464022007086209, 7.945255867625545287452307043017, 8.671013120397769676899477391490