L(s) = 1 | − 2.46·2-s + 4.05·4-s + (−1.82 − 3.16i)5-s − 5.05·8-s + (4.50 + 7.79i)10-s + (0.203 − 0.351i)11-s + (0.243 − 0.421i)13-s + 4.32·16-s + (2.42 + 4.20i)17-s + (0.986 − 1.70i)19-s + (−7.41 − 12.8i)20-s + (−0.5 + 0.866i)22-s + (2.32 + 4.02i)23-s + (−4.19 + 7.25i)25-s + (−0.598 + 1.03i)26-s + ⋯ |
L(s) = 1 | − 1.73·2-s + 2.02·4-s + (−0.817 − 1.41i)5-s − 1.78·8-s + (1.42 + 2.46i)10-s + (0.0612 − 0.106i)11-s + (0.0675 − 0.116i)13-s + 1.08·16-s + (0.588 + 1.01i)17-s + (0.226 − 0.392i)19-s + (−1.65 − 2.87i)20-s + (−0.106 + 0.184i)22-s + (0.484 + 0.839i)23-s + (−0.838 + 1.45i)25-s + (−0.117 + 0.203i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5940410367\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5940410367\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 5 | \( 1 + (1.82 + 3.16i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.203 + 0.351i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.243 + 0.421i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.42 - 4.20i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.986 + 1.70i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.32 - 4.02i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.82 - 6.62i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 + (1.16 - 2.01i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.75 + 6.50i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.16 - 2.01i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6.31T + 47T^{2} \) |
| 53 | \( 1 + (1.78 + 3.09i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 6.11T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 - 3.60T + 67T^{2} \) |
| 71 | \( 1 + 8.46T + 71T^{2} \) |
| 73 | \( 1 + (0.986 + 1.70i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 8.16T + 79T^{2} \) |
| 83 | \( 1 + (-6.08 - 10.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.41 + 12.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.74 - 8.21i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.165860564865015270840239530975, −8.864115057472261874704374874123, −8.043236528300539195994432326499, −7.61601272953496087011284455057, −6.58089463178952441647479530431, −5.44604929935622401441513242510, −4.41207707529591522441895164539, −3.16376671796952848201906829704, −1.54789674998938398771354830822, −0.74346174352306033589645740506,
0.76341914505706449210158371551, 2.44165767051251130760514809354, 3.11494114235324957303412933576, 4.47181133495484690327980358920, 6.17193309644191933255635344889, 6.75660061059189716999409479639, 7.63621634385719331881944066000, 7.893039689443258085162152047261, 8.958739317588466764678731280828, 9.765799488373076647358086207542