| L(s) = 1 | + (−2.48 + 14.0i)2-s + (1.73e3 + 630. i)4-s + (8.75e3 − 7.34e3i)5-s + (−7.41e4 + 2.69e4i)7-s + (−2.78e4 + 4.81e4i)8-s + (8.16e4 + 1.41e5i)10-s + (−3.46e5 − 2.90e5i)11-s + (4.03e4 + 2.28e5i)13-s + (−1.95e5 − 1.11e6i)14-s + (2.28e6 + 1.91e6i)16-s + (8.69e4 + 1.50e5i)17-s + (7.81e6 − 1.35e7i)19-s + (1.97e7 − 7.20e6i)20-s + (4.94e6 − 4.15e6i)22-s + (3.55e7 + 1.29e7i)23-s + ⋯ |
| L(s) = 1 | + (−0.0548 + 0.311i)2-s + (0.845 + 0.307i)4-s + (1.25 − 1.05i)5-s + (−1.66 + 0.606i)7-s + (−0.300 + 0.519i)8-s + (0.258 + 0.447i)10-s + (−0.647 − 0.543i)11-s + (0.0301 + 0.171i)13-s + (−0.0973 − 0.552i)14-s + (0.544 + 0.456i)16-s + (0.0148 + 0.0257i)17-s + (0.723 − 1.25i)19-s + (1.38 − 0.503i)20-s + (0.204 − 0.171i)22-s + (1.15 + 0.419i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.01588 - 1.02544i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.01588 - 1.02544i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (2.48 - 14.0i)T + (-1.92e3 - 700. i)T^{2} \) |
| 5 | \( 1 + (-8.75e3 + 7.34e3i)T + (8.47e6 - 4.80e7i)T^{2} \) |
| 7 | \( 1 + (7.41e4 - 2.69e4i)T + (1.51e9 - 1.27e9i)T^{2} \) |
| 11 | \( 1 + (3.46e5 + 2.90e5i)T + (4.95e10 + 2.80e11i)T^{2} \) |
| 13 | \( 1 + (-4.03e4 - 2.28e5i)T + (-1.68e12 + 6.12e11i)T^{2} \) |
| 17 | \( 1 + (-8.69e4 - 1.50e5i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-7.81e6 + 1.35e7i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-3.55e7 - 1.29e7i)T + (7.29e14 + 6.12e14i)T^{2} \) |
| 29 | \( 1 + (-1.17e7 + 6.66e7i)T + (-1.14e16 - 4.17e15i)T^{2} \) |
| 31 | \( 1 + (1.51e7 + 5.50e6i)T + (1.94e16 + 1.63e16i)T^{2} \) |
| 37 | \( 1 + (4.03e8 + 6.98e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + (1.15e8 + 6.53e8i)T + (-5.17e17 + 1.88e17i)T^{2} \) |
| 43 | \( 1 + (-6.60e8 - 5.54e8i)T + (1.61e17 + 9.15e17i)T^{2} \) |
| 47 | \( 1 + (-2.58e9 + 9.40e8i)T + (1.89e18 - 1.58e18i)T^{2} \) |
| 53 | \( 1 + 3.14e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-1.09e9 + 9.22e8i)T + (5.23e18 - 2.96e19i)T^{2} \) |
| 61 | \( 1 + (9.23e8 - 3.36e8i)T + (3.33e19 - 2.79e19i)T^{2} \) |
| 67 | \( 1 + (2.66e9 + 1.51e10i)T + (-1.14e20 + 4.17e19i)T^{2} \) |
| 71 | \( 1 + (2.55e9 + 4.42e9i)T + (-1.15e20 + 2.00e20i)T^{2} \) |
| 73 | \( 1 + (-6.02e8 + 1.04e9i)T + (-1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-3.10e9 + 1.75e10i)T + (-7.02e20 - 2.55e20i)T^{2} \) |
| 83 | \( 1 + (1.42e9 - 8.06e9i)T + (-1.21e21 - 4.40e20i)T^{2} \) |
| 89 | \( 1 + (-1.31e10 + 2.27e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + (6.33e10 + 5.31e10i)T + (1.24e21 + 7.04e21i)T^{2} \) |
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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
−12.23380417905670372622423014587, −10.75899778409385151301597453079, −9.418702475878874670270223920212, −8.870589167544759280219475159748, −7.16636458856659135472031931145, −6.03987713904663153214886986202, −5.36133840805305543618382678330, −3.13287140308460314824565645219, −2.19614411369261321654210296337, −0.54768747405403747831438608124,
1.24069292198396646594498312602, 2.63859817388306007152716424772, 3.27056371998199097111510627196, 5.65553314906963068555483771821, 6.57842818054512980023637694026, 7.23750028819828216866559951762, 9.560685583366919305594745872663, 10.19464588405610947265693690806, 10.71243346936336890994489135676, 12.35055511627410209146316419060
