| 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
−11.38113852686060478076855866333, −10.40985154604808611784273185243, −9.296711674382182081515018626793, −8.454736433108972456077023656957, −6.64373375289584375693292504289, −5.64369261207424976177169939518, −4.47983462721713936296224252112, −2.43118371575981034936896366574, −1.53747917507644610539678387536, −0.26221400082951661233916303390,
1.76928376463429465527676446153, 3.01632242329721937817478197536, 4.27632348919143993244582538569, 6.33932383935758986055138587182, 6.79622498777216274115081329868, 8.185340375380579471893024930552, 9.184602602765877171509345821173, 10.82425953396835036252086054146, 11.30884774950397329432759493979, 12.76731950721327062770868862036
