| L(s) = 1 | + (7.27 − 8.66i)2-s + (−62.6 − 51.3i)3-s + (−22.2 − 126. i)4-s + (−16.8 + 46.2i)5-s + (−900. + 169. i)6-s + (−338. + 1.91e3i)7-s + (−1.25e3 − 724. i)8-s + (1.28e3 + 6.43e3i)9-s + (278. + 482. i)10-s + (2.87e3 + 7.89e3i)11-s + (−5.07e3 + 9.03e3i)12-s + (2.16e3 − 1.81e3i)13-s + (1.41e4 + 1.68e4i)14-s + (3.43e3 − 2.03e3i)15-s + (−1.53e4 + 5.60e3i)16-s + (5.03e3 − 2.90e3i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (−0.773 − 0.633i)3-s + (−0.0868 − 0.492i)4-s + (−0.0269 + 0.0740i)5-s + (−0.694 + 0.130i)6-s + (−0.140 + 0.798i)7-s + (−0.306 − 0.176i)8-s + (0.196 + 0.980i)9-s + (0.0278 + 0.0482i)10-s + (0.196 + 0.539i)11-s + (−0.244 + 0.435i)12-s + (0.0756 − 0.0634i)13-s + (0.368 + 0.439i)14-s + (0.0677 − 0.0401i)15-s + (−0.234 + 0.0855i)16-s + (0.0603 − 0.0348i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0235i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.999 + 0.0235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.58596 - 0.0186697i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58596 - 0.0186697i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.27 + 8.66i)T \) |
| 3 | \( 1 + (62.6 + 51.3i)T \) |
| good | 5 | \( 1 + (16.8 - 46.2i)T + (-2.99e5 - 2.51e5i)T^{2} \) |
| 7 | \( 1 + (338. - 1.91e3i)T + (-5.41e6 - 1.97e6i)T^{2} \) |
| 11 | \( 1 + (-2.87e3 - 7.89e3i)T + (-1.64e8 + 1.37e8i)T^{2} \) |
| 13 | \( 1 + (-2.16e3 + 1.81e3i)T + (1.41e8 - 8.03e8i)T^{2} \) |
| 17 | \( 1 + (-5.03e3 + 2.90e3i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.57e4 + 2.72e4i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-2.47e5 + 4.35e4i)T + (7.35e10 - 2.67e10i)T^{2} \) |
| 29 | \( 1 + (-9.03e4 + 1.07e5i)T + (-8.68e10 - 4.92e11i)T^{2} \) |
| 31 | \( 1 + (-1.05e5 - 6.00e5i)T + (-8.01e11 + 2.91e11i)T^{2} \) |
| 37 | \( 1 + (2.71e5 + 4.70e5i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-2.33e6 - 2.78e6i)T + (-1.38e12 + 7.86e12i)T^{2} \) |
| 43 | \( 1 + (1.89e6 - 6.88e5i)T + (8.95e12 - 7.51e12i)T^{2} \) |
| 47 | \( 1 + (-4.29e6 - 7.56e5i)T + (2.23e13 + 8.14e12i)T^{2} \) |
| 53 | \( 1 - 6.92e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (2.87e6 - 7.89e6i)T + (-1.12e14 - 9.43e13i)T^{2} \) |
| 61 | \( 1 + (8.97e5 - 5.08e6i)T + (-1.80e14 - 6.55e13i)T^{2} \) |
| 67 | \( 1 + (7.33e6 - 6.15e6i)T + (7.05e13 - 3.99e14i)T^{2} \) |
| 71 | \( 1 + (3.70e7 - 2.14e7i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-4.34e6 + 7.53e6i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (1.83e7 + 1.53e7i)T + (2.63e14 + 1.49e15i)T^{2} \) |
| 83 | \( 1 + (2.80e7 - 3.34e7i)T + (-3.91e14 - 2.21e15i)T^{2} \) |
| 89 | \( 1 + (-4.04e7 - 2.33e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-1.63e8 + 5.93e7i)T + (6.00e15 - 5.03e15i)T^{2} \) |
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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
−13.24856638917546187239458819495, −12.43481850028431120849838551987, −11.56496686057832037320291779324, −10.48428428824889997627661949444, −8.996420639817091273325068540856, −7.20788362442616895832055704192, −5.93663894137512683224716773967, −4.75641108453998048807890068349, −2.70113157080928010768803835933, −1.19316461607643201811156716483,
0.62344476284190899189837995034, 3.50840148713300754211641459819, 4.70004214589255685936113571523, 6.01017040958373209977743979391, 7.17441220365641315180119952726, 8.853011672393998128504254334617, 10.27307405870296940434062150119, 11.33531825340371378793672916150, 12.53821569961732281006182966238, 13.74917448212341660097263909575
