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
\(L(1)\) |
\(\approx\) |
\(0.5651277774\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5651277774\) |
\(L(\frac{3}{2})\) |
|
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
−9.873799823096550113654212999387, −9.290188938328988779574991626499, −8.491934205997079343626232495561, −7.45938187605495455290911758781, −6.88849848703584986445332440678, −6.35943623882130171861546341056, −5.18360878848245287746151197403, −3.33134772842876584367018283957, −2.48467502629714309032843095202, −1.49931259116447002012373247851,
0.41282626173249497501780919409, 1.58417748683689928144700318186, 2.37623066443111690723583729686, 4.20675792631025688991338487847, 5.29792276352109546703460381098, 6.24425928352753514699912582102, 7.04594486678233568742716469434, 8.176835502305385076290677514402, 8.631107646654688792307234413083, 9.184433062620140668493778377556