L(s) = 1 | + (1.47 − 2.55i)2-s + (11.6 + 20.1i)4-s + (32.4 + 56.1i)5-s + (80.0 − 138. i)7-s + 163.·8-s + 191.·10-s + (−103. + 178. i)11-s + (32.8 + 56.9i)13-s + (−236. − 409. i)14-s + (−131. + 227. i)16-s + 601.·17-s − 2.00e3·19-s + (−754. + 1.30e3i)20-s + (304. + 528. i)22-s + (−2.18e3 − 3.78e3i)23-s + ⋯ |
L(s) = 1 | + (0.261 − 0.452i)2-s + (0.363 + 0.629i)4-s + (0.580 + 1.00i)5-s + (0.617 − 1.07i)7-s + 0.901·8-s + 0.605·10-s + (−0.257 + 0.445i)11-s + (0.0539 + 0.0934i)13-s + (−0.322 − 0.558i)14-s + (−0.128 + 0.222i)16-s + 0.504·17-s − 1.27·19-s + (−0.421 + 0.730i)20-s + (0.134 + 0.232i)22-s + (−0.861 − 1.49i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.03231 + 0.116974i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.03231 + 0.116974i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (-1.47 + 2.55i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-32.4 - 56.1i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-80.0 + 138. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (103. - 178. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-32.8 - 56.9i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 - 601.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.00e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (2.18e3 + 3.78e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (467. - 809. i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (2.62e3 + 4.54e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 5.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (3.22e3 + 5.58e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.40e3 + 2.43e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-3.90e3 + 6.76e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + 3.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-7.84e3 - 1.35e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.11e4 - 1.93e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-9.45e3 - 1.63e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 4.67e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.53e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-1.56e4 + 2.70e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (3.09e4 - 5.35e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 - 1.21e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (3.67e4 - 6.36e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−16.55898905040198178595031938177, −14.78150760971475461250476571801, −13.80762469178097728437153218149, −12.53386323626020230988443972224, −10.98598244161835331364415768637, −10.30521379601396833998772235960, −7.923380027866324995016681955924, −6.67398525535259677539452475009, −4.15443477876413838454745383529, −2.27037253714779250498393950444,
1.72297001227169356495419369837, 5.07768881095233666089705061209, 5.98908495764928910896893669677, 8.144868168603859136004584506927, 9.538878578546512081613044876212, 11.16499080387105510116208328396, 12.63304754041091707559956002360, 13.94618357193520336457852341225, 15.11750075819191628805799683920, 16.11511141254610028379014701594