L(s) = 1 | − 2.53·2-s + 4.41·4-s + (−0.439 − 0.761i)5-s − 6.10·8-s + (1.11 + 1.92i)10-s + (1.93 − 3.35i)11-s + (2.72 − 4.72i)13-s + 6.63·16-s + (0.826 + 1.43i)17-s + (−1.20 + 2.08i)19-s + (−1.93 − 3.35i)20-s + (−4.91 + 8.50i)22-s + (1.58 + 2.73i)23-s + (2.11 − 3.66i)25-s + (−6.90 + 11.9i)26-s + ⋯ |
L(s) = 1 | − 1.79·2-s + 2.20·4-s + (−0.196 − 0.340i)5-s − 2.15·8-s + (0.352 + 0.609i)10-s + (0.584 − 1.01i)11-s + (0.756 − 1.30i)13-s + 1.65·16-s + (0.200 + 0.347i)17-s + (−0.276 + 0.479i)19-s + (−0.433 − 0.751i)20-s + (−1.04 + 1.81i)22-s + (0.329 + 0.571i)23-s + (0.422 − 0.732i)25-s + (−1.35 + 2.34i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5779825095\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5779825095\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.53T + 2T^{2} \) |
| 5 | \( 1 + (0.439 + 0.761i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.93 + 3.35i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.72 + 4.72i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.826 - 1.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.20 - 2.08i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.58 - 2.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.02 + 5.23i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4.55T + 31T^{2} \) |
| 37 | \( 1 + (-2.27 + 3.94i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.592 - 1.02i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.0923 + 0.160i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 1.02T + 47T^{2} \) |
| 53 | \( 1 + (-3.64 - 6.31i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 6.66T + 59T^{2} \) |
| 61 | \( 1 + 2.59T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + (-6.39 - 11.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 5.95T + 79T^{2} \) |
| 83 | \( 1 + (0.109 + 0.189i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.51 + 9.54i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.25 + 10.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.244116223205323892680879734360, −8.555659958050659060895964195459, −8.077338752289922858293399190330, −7.33462126652440734100162831030, −6.19889802127740087079879918679, −5.66912255894960011302937437285, −3.93292738038095162567008306760, −2.89097028496577492197810740024, −1.47004014590917633133735004636, −0.48897064206948034199398703299,
1.30529625134466964710113738245, 2.21192925779734216232361606587, 3.54060668688092052242506071789, 4.79542064840869391124347659342, 6.31287846088186572433727937419, 6.95760180161836737638631825652, 7.39672399764995333554511284886, 8.500095630900863322051229757022, 9.174000388481951748698729741493, 9.543124114385991004877208100455