| L(s) = 1 | + (18.2 + 3.22i)2-s + (83.2 + 30.2i)4-s + (223. + 266. i)5-s + (−1.43e3 + 523. i)7-s + (−2.69e3 − 1.55e3i)8-s + (3.22e3 + 5.58e3i)10-s + (−262. + 312. i)11-s + (−7.24e3 − 4.10e4i)13-s + (−2.79e4 + 4.93e3i)14-s + (−6.15e4 − 5.16e4i)16-s + (−5.26e4 + 3.03e4i)17-s + (8.16e4 − 1.41e5i)19-s + (1.05e4 + 2.89e4i)20-s + (−5.80e3 + 4.87e3i)22-s + (−8.85e4 + 2.43e5i)23-s + ⋯ |
| L(s) = 1 | + (1.14 + 0.201i)2-s + (0.325 + 0.118i)4-s + (0.357 + 0.425i)5-s + (−0.599 + 0.218i)7-s + (−0.657 − 0.379i)8-s + (0.322 + 0.558i)10-s + (−0.0179 + 0.0213i)11-s + (−0.253 − 1.43i)13-s + (−0.728 + 0.128i)14-s + (−0.939 − 0.788i)16-s + (−0.630 + 0.363i)17-s + (0.626 − 1.08i)19-s + (0.0657 + 0.180i)20-s + (−0.0247 + 0.0207i)22-s + (−0.316 + 0.869i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.813i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.580 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.531844 - 1.03301i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.531844 - 1.03301i\) |
| \(L(5)\) |
|
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 | \( 1 + (-18.2 - 3.22i)T + (240. + 87.5i)T^{2} \) |
| 5 | \( 1 + (-223. - 266. i)T + (-6.78e4 + 3.84e5i)T^{2} \) |
| 7 | \( 1 + (1.43e3 - 523. i)T + (4.41e6 - 3.70e6i)T^{2} \) |
| 11 | \( 1 + (262. - 312. i)T + (-3.72e7 - 2.11e8i)T^{2} \) |
| 13 | \( 1 + (7.24e3 + 4.10e4i)T + (-7.66e8 + 2.78e8i)T^{2} \) |
| 17 | \( 1 + (5.26e4 - 3.03e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-8.16e4 + 1.41e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (8.85e4 - 2.43e5i)T + (-5.99e10 - 5.03e10i)T^{2} \) |
| 29 | \( 1 + (8.33e5 + 1.46e5i)T + (4.70e11 + 1.71e11i)T^{2} \) |
| 31 | \( 1 + (7.33e5 + 2.67e5i)T + (6.53e11 + 5.48e11i)T^{2} \) |
| 37 | \( 1 + (8.58e5 + 1.48e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-2.46e4 + 4.34e3i)T + (7.50e12 - 2.73e12i)T^{2} \) |
| 43 | \( 1 + (-7.38e5 - 6.20e5i)T + (2.02e12 + 1.15e13i)T^{2} \) |
| 47 | \( 1 + (4.63e5 + 1.27e6i)T + (-1.82e13 + 1.53e13i)T^{2} \) |
| 53 | \( 1 + 6.06e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-1.23e7 - 1.47e7i)T + (-2.54e13 + 1.44e14i)T^{2} \) |
| 61 | \( 1 + (-7.35e6 + 2.67e6i)T + (1.46e14 - 1.23e14i)T^{2} \) |
| 67 | \( 1 + (3.57e6 + 2.02e7i)T + (-3.81e14 + 1.38e14i)T^{2} \) |
| 71 | \( 1 + (1.86e7 - 1.07e7i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (2.63e7 - 4.56e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-6.77e6 + 3.84e7i)T + (-1.42e15 - 5.18e14i)T^{2} \) |
| 83 | \( 1 + (-1.82e7 - 3.21e6i)T + (2.11e15 + 7.70e14i)T^{2} \) |
| 89 | \( 1 + (7.85e7 + 4.53e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-5.90e7 - 4.95e7i)T + (1.36e15 + 7.71e15i)T^{2} \) |
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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
−12.83244760892479008448594664651, −11.48624757352698015593416781150, −10.14106374086795728900723386938, −9.078826106665860333349429402120, −7.32908477948032855850180397827, −6.09015055048640910550799002332, −5.23865186374708953098259183096, −3.68440563490981270916276798134, −2.58562421751987839316282069261, −0.22073330481284153406783500844,
1.89536314434532422866142720314, 3.46454413084602783665316781666, 4.57923008337421304434263044706, 5.75305354212420995963997130700, 6.96532383811445151554972616942, 8.793852092970242678830641544339, 9.704062597169782834025534502130, 11.29585760752933070812571485391, 12.28167255059503283860985564945, 13.13830374214423571265419364913
