| L(s) = 1 | + (−8.23 + 14.2i)2-s + (888. + 1.53e3i)4-s + (3.32e3 + 5.76e3i)5-s + (1.18e4 − 2.05e4i)7-s − 6.29e4·8-s − 1.09e5·10-s + (1.82e5 − 3.15e5i)11-s + (6.03e5 + 1.04e6i)13-s + (1.95e5 + 3.38e5i)14-s + (−1.30e6 + 2.25e6i)16-s − 6.85e6·17-s − 1.57e7·19-s + (−5.91e6 + 1.02e7i)20-s + (3.00e6 + 5.19e6i)22-s + (9.64e6 + 1.67e7i)23-s + ⋯ |
| L(s) = 1 | + (−0.181 + 0.315i)2-s + (0.433 + 0.751i)4-s + (0.476 + 0.825i)5-s + (0.266 − 0.461i)7-s − 0.679·8-s − 0.346·10-s + (0.341 − 0.590i)11-s + (0.450 + 0.780i)13-s + (0.0970 + 0.168i)14-s + (−0.310 + 0.537i)16-s − 1.17·17-s − 1.45·19-s + (−0.413 + 0.715i)20-s + (0.124 + 0.215i)22-s + (0.312 + 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.192611 - 1.09235i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.192611 - 1.09235i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (8.23 - 14.2i)T + (-1.02e3 - 1.77e3i)T^{2} \) |
| 5 | \( 1 + (-3.32e3 - 5.76e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 7 | \( 1 + (-1.18e4 + 2.05e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-1.82e5 + 3.15e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + (-6.03e5 - 1.04e6i)T + (-8.96e11 + 1.55e12i)T^{2} \) |
| 17 | \( 1 + 6.85e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.57e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + (-9.64e6 - 1.67e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (4.08e7 - 7.07e7i)T + (-6.10e15 - 1.05e16i)T^{2} \) |
| 31 | \( 1 + (-5.31e7 - 9.21e7i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 - 2.75e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + (1.61e8 + 2.79e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (7.17e8 - 1.24e9i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + (7.73e7 - 1.34e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + 4.70e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (2.63e9 + 4.56e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-2.01e9 + 3.49e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (2.06e9 + 3.57e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 2.18e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.70e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + (1.80e10 - 3.13e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + (-2.86e10 + 4.97e10i)T + (-6.43e20 - 1.11e21i)T^{2} \) |
| 89 | \( 1 + 1.04e11T + 2.77e21T^{2} \) |
| 97 | \( 1 + (2.32e10 - 4.02e10i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
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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
−12.76731950721327062770868862036, −11.30884774950397329432759493979, −10.82425953396835036252086054146, −9.184602602765877171509345821173, −8.185340375380579471893024930552, −6.79622498777216274115081329868, −6.33932383935758986055138587182, −4.27632348919143993244582538569, −3.01632242329721937817478197536, −1.76928376463429465527676446153,
0.26221400082951661233916303390, 1.53747917507644610539678387536, 2.43118371575981034936896366574, 4.47983462721713936296224252112, 5.64369261207424976177169939518, 6.64373375289584375693292504289, 8.454736433108972456077023656957, 9.296711674382182081515018626793, 10.40985154604808611784273185243, 11.38113852686060478076855866333
