| L(s) = 1 | + (−8.29 − 14.3i)2-s + (−73.6 + 127. i)4-s + (−57.0 + 98.8i)5-s + (−719. − 1.24e3i)7-s + 320.·8-s + 1.89e3·10-s + (−2.96e3 − 5.13e3i)11-s + (5.72e3 − 9.91e3i)13-s + (−1.19e4 + 2.06e4i)14-s + (6.76e3 + 1.17e4i)16-s − 2.02e4·17-s − 6.35e3·19-s + (−8.41e3 − 1.45e4i)20-s + (−4.91e4 + 8.51e4i)22-s + (−3.79e4 + 6.56e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.733 − 1.27i)2-s + (−0.575 + 0.996i)4-s + (−0.204 + 0.353i)5-s + (−0.792 − 1.37i)7-s + 0.221·8-s + 0.599·10-s + (−0.671 − 1.16i)11-s + (0.722 − 1.25i)13-s + (−1.16 + 2.01i)14-s + (0.413 + 0.715i)16-s − 0.998·17-s − 0.212·19-s + (−0.235 − 0.407i)20-s + (−0.984 + 1.70i)22-s + (−0.649 + 1.12i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0386569 + 0.0140699i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0386569 + 0.0140699i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (8.29 + 14.3i)T + (-64 + 110. i)T^{2} \) |
| 5 | \( 1 + (57.0 - 98.8i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (719. + 1.24e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (2.96e3 + 5.13e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-5.72e3 + 9.91e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + 2.02e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 6.35e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + (3.79e4 - 6.56e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-3.73e4 - 6.47e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-9.46e4 + 1.63e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.34e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (7.06e4 - 1.22e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.23e5 - 2.13e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (1.67e5 + 2.90e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 - 1.65e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.02e6 - 1.77e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.95e5 - 5.11e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.67e4 - 4.63e4i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 4.95e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 8.17e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (3.78e6 + 6.55e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-5.09e5 - 8.82e5i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 1.37e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-5.30e6 - 9.18e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−11.51219813011226259187958313851, −10.67782826018809192787803220272, −10.18269509260099095272819910127, −8.787122750312978573654394421954, −7.61348390936002952912783789080, −6.04961476349489908367202594276, −3.75037071180836264091732757065, −2.93900583060122940847668151462, −0.977647064736156044238460879109, −0.02152337680190510211394280358,
2.35653300627006348107225912953, 4.64865420775663160489088298364, 6.12718822591630372598392694069, 6.88156402191491428397361319870, 8.451383896729531522942746342209, 8.965690873889495235492535145981, 10.09917307467967173422658999573, 11.93094928434930780493120963702, 12.73077347947956133749640318174, 14.24337430094005766153934928411
