| L(s) = 1 | + (−3.77 + 6.53i)2-s + (−15.5 − 0.0716i)3-s + (−12.4 − 21.6i)4-s + (43.7 + 75.7i)5-s + (59.3 − 101. i)6-s + (20.6 − 35.6i)7-s − 52.9·8-s + (242. + 2.23i)9-s − 660.·10-s + (3.84 − 6.65i)11-s + (193. + 338. i)12-s + (302. + 524. i)13-s + (155. + 269. i)14-s + (−676. − 1.18e3i)15-s + (599. − 1.03e3i)16-s − 566.·17-s + ⋯ |
| L(s) = 1 | + (−0.667 + 1.15i)2-s + (−0.999 − 0.00459i)3-s + (−0.390 − 0.676i)4-s + (0.782 + 1.35i)5-s + (0.672 − 1.15i)6-s + (0.158 − 0.275i)7-s − 0.292·8-s + (0.999 + 0.00919i)9-s − 2.08·10-s + (0.00957 − 0.0165i)11-s + (0.387 + 0.678i)12-s + (0.496 + 0.860i)13-s + (0.212 + 0.367i)14-s + (−0.775 − 1.35i)15-s + (0.585 − 1.01i)16-s − 0.475·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.235693 + 0.638411i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.235693 + 0.638411i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (15.5 + 0.0716i)T \) |
| good | 2 | \( 1 + (3.77 - 6.53i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-43.7 - 75.7i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-20.6 + 35.6i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-3.84 + 6.65i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-302. - 524. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + 566.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.06e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-137. - 237. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-2.41e3 + 4.17e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (2.23e3 + 3.86e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 242.T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-1.75e3 - 3.03e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (4.26e3 - 7.39e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-8.99e3 + 1.55e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 - 2.02e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (1.00e3 + 1.74e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.41e4 + 2.45e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.09e4 + 3.62e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 3.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (4.40e4 - 7.63e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-5.28e3 + 9.16e3i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + 1.24e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-8.14e4 + 1.41e5i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−21.34888866552834209133817088295, −18.60883715777978717945175567515, −17.93760901312571844170937089252, −16.84348563596771612958474948017, −15.49868153593131324392784291612, −13.93644953649117869724442963581, −11.36822912785953947395130531502, −9.774204504890603386087452986914, −7.17726103821870938640173135940, −6.06886734385210333074671945691,
1.13686792571298347011876015125, 5.47533151566393401961020912824, 8.984052524867929766769366472853, 10.39730576517536898384740887354, 11.95115158941177323876078034490, 13.03947156962472926109989851878, 15.99888460285536846505986576839, 17.48183657656863750384143471526, 18.23985152688004655150275736308, 20.09118508814273339627133375901
