L(s) = 1 | + (1.66 − 2.87i)5-s − 2.32·7-s − 1.70·11-s + (−2.01 − 3.48i)13-s + (−0.193 − 4.35i)19-s + (1.17 + 2.03i)23-s + (−3.01 − 5.22i)25-s + (3.32 + 5.75i)29-s − 6.70·31-s + (−3.85 + 6.67i)35-s − 37-s + (−3.32 + 5.75i)41-s + (0.353 − 0.612i)43-s + (3 + 5.19i)47-s − 1.61·49-s + ⋯ |
L(s) = 1 | + (0.742 − 1.28i)5-s − 0.877·7-s − 0.514·11-s + (−0.558 − 0.967i)13-s + (−0.0443 − 0.999i)19-s + (0.244 + 0.424i)23-s + (−0.602 − 1.04i)25-s + (0.616 + 1.06i)29-s − 1.20·31-s + (−0.651 + 1.12i)35-s − 0.164·37-s + (−0.518 + 0.898i)41-s + (0.0539 − 0.0934i)43-s + (0.437 + 0.757i)47-s − 0.230·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5487550262\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5487550262\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (0.193 + 4.35i)T \) |
good | 5 | \( 1 + (-1.66 + 2.87i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 2.32T + 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 + (2.01 + 3.48i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-1.17 - 2.03i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.32 - 5.75i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (3.32 - 5.75i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.353 + 0.612i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.98 + 8.62i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.853 - 1.47i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.69 + 2.93i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.18 + 7.25i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.70 + 8.15i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.82 - 10.0i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.67 - 2.90i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + (1.33 + 2.32i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.86 - 15.3i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.565557486848183820950380302047, −7.72108863396445674977483234755, −6.85932232905582628178199909535, −5.98370804802411753095379014859, −5.16011594590374936385166845908, −4.82720346680231829109173373973, −3.43716241866183184997895035283, −2.61666538947727443858617402679, −1.37783453169289343612390786265, −0.16713998856345718772603329228,
1.88451397674878543224435337605, 2.65951122310780824906612709034, 3.45290698426526208624439960836, 4.45202662315622975132039704083, 5.67765612549069325930901727930, 6.18310792152213182994670578116, 6.97093534771182888983278128057, 7.42425879171264075781425612686, 8.566000663654990794940681227938, 9.417669481621027151513885234542