| L(s) = 1 | + (−12 − 20.7i)2-s + (736 − 1.27e3i)4-s + (2.41e3 − 4.18e3i)5-s + (8.37e3 + 1.45e4i)7-s − 8.44e4·8-s − 1.15e5·10-s + (2.67e5 + 4.62e5i)11-s + (2.88e5 − 5.00e5i)13-s + (2.00e5 − 3.48e5i)14-s + (−4.93e5 − 8.54e5i)16-s + 6.90e6·17-s + 1.06e7·19-s + (−3.55e6 − 6.15e6i)20-s + (6.41e6 − 1.11e7i)22-s + (9.32e6 − 1.61e7i)23-s + ⋯ |
| L(s) = 1 | + (−0.265 − 0.459i)2-s + (0.359 − 0.622i)4-s + (0.345 − 0.598i)5-s + (0.188 + 0.326i)7-s − 0.911·8-s − 0.366·10-s + (0.500 + 0.866i)11-s + (0.215 − 0.373i)13-s + (0.0998 − 0.172i)14-s + (−0.117 − 0.203i)16-s + 1.17·17-s + 0.987·19-s + (−0.248 − 0.430i)20-s + (0.265 − 0.459i)22-s + (0.301 − 0.523i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.89761 - 1.59229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89761 - 1.59229i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (12 + 20.7i)T + (-1.02e3 + 1.77e3i)T^{2} \) |
| 5 | \( 1 + (-2.41e3 + 4.18e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 7 | \( 1 + (-8.37e3 - 1.45e4i)T + (-9.88e8 + 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-2.67e5 - 4.62e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + (-2.88e5 + 5.00e5i)T + (-8.96e11 - 1.55e12i)T^{2} \) |
| 17 | \( 1 - 6.90e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.06e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + (-9.32e6 + 1.61e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + (-6.42e7 - 1.11e8i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 + (-2.64e7 + 4.57e7i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + 1.82e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + (-1.54e8 + 2.66e8i)T + (-2.75e17 - 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-8.56e6 - 1.48e7i)T + (-4.64e17 + 8.04e17i)T^{2} \) |
| 47 | \( 1 + (-1.34e9 - 2.32e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 - 1.59e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (2.59e9 - 4.49e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (3.47e9 + 6.02e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-7.74e9 + 1.34e10i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + 9.79e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.46e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + (1.90e10 + 3.30e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + (1.46e10 + 2.54e10i)T + (-6.43e20 + 1.11e21i)T^{2} \) |
| 89 | \( 1 - 2.49e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (3.75e10 + 6.49e10i)T + (-3.57e21 + 6.19e21i)T^{2} \) |
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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
−11.91327912242625468932796993124, −10.65230092226186076017476356539, −9.689099109317185505522958231589, −8.861444623648992809362197922197, −7.29197195718876774025270062624, −5.88282563065608915340695982593, −4.94111835572210457713655692994, −3.07256652303548237010034264524, −1.66325993507941075389011032192, −0.869690262287761992245214114731,
1.00791905800247332788049437725, 2.73514293262995343029826439398, 3.75457977162977610100818902895, 5.70328837789708976202407376700, 6.74454854572230267078085640248, 7.72384641226819788689449230518, 8.822002870826930325942276110398, 10.09357525847355090324399601633, 11.35325924775946434612992574627, 12.12794055064635834188343144503
