L(s) = 1 | + (0.119 − 0.207i)2-s + (0.971 + 1.68i)4-s + 2.59·5-s + 0.942·8-s + (0.309 − 0.536i)10-s − 4.18·11-s + (−1.84 + 3.18i)13-s + (−1.83 + 3.16i)16-s + (0.855 − 1.48i)17-s + (3.57 + 6.19i)19-s + (2.51 + 4.36i)20-s + (−0.499 + 0.866i)22-s + 5.12·23-s + 1.71·25-s + (0.440 + 0.762i)26-s + ⋯ |
L(s) = 1 | + (0.0845 − 0.146i)2-s + (0.485 + 0.841i)4-s + 1.15·5-s + 0.333·8-s + (0.0979 − 0.169i)10-s − 1.26·11-s + (−0.510 + 0.884i)13-s + (−0.457 + 0.792i)16-s + (0.207 − 0.359i)17-s + (0.820 + 1.42i)19-s + (0.562 + 0.975i)20-s + (−0.106 + 0.184i)22-s + 1.06·23-s + 0.343·25-s + (0.0863 + 0.149i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.203072267\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.203072267\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.119 + 0.207i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 2.59T + 5T^{2} \) |
| 11 | \( 1 + 4.18T + 11T^{2} \) |
| 13 | \( 1 + (1.84 - 3.18i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.855 + 1.48i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.57 - 6.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 + (1.06 + 1.84i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.26 - 5.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.830 + 1.43i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.10 - 8.84i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.830 - 1.43i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.66 + 8.08i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.32 + 9.22i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.03 - 5.25i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.99 + 6.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.13 + 7.15i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6.23T + 71T^{2} \) |
| 73 | \( 1 + (3.57 - 6.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.91 + 8.51i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.44 - 5.97i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.51 + 4.36i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.53 + 2.65i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.975352744375175042204702101185, −8.993364082761331689508559696108, −8.095264702106197470275971071813, −7.33333248899498786882541659326, −6.57168459880016558721629007504, −5.55100224751857458245569097347, −4.81086665852899545704645874742, −3.46045355546798776187721945798, −2.57913835387543878902828257225, −1.69568617851227494755837155947,
0.870409223148638665232873813034, 2.28474505584235212381072481845, 2.89942236397684887495704802629, 4.79013565417520332287376365555, 5.44152088307267919857078606045, 5.90581160126552098881566737568, 7.04627509557623198147241369173, 7.59598871910710869978371548285, 8.864831538988767627386949528938, 9.638628170883709943683794467931