| L(s) = 1 | − 41.6i·2-s + (347. − 200. i)3-s − 712.·4-s + (−2.81e3 + 1.62e3i)5-s + (−8.35e3 − 1.44e4i)6-s + (−3.43e3 − 1.98e3i)7-s − 1.29e4i·8-s + (5.08e4 − 8.81e4i)9-s + (6.76e4 + 1.17e5i)10-s − 1.94e5·11-s + (−2.47e5 + 1.42e5i)12-s + (−1.62e5 + 2.81e5i)13-s + (−8.25e4 + 1.42e5i)14-s + (−6.51e5 + 1.12e6i)15-s − 1.27e6·16-s + (5.06e5 − 8.77e5i)17-s + ⋯ |
| L(s) = 1 | − 1.30i·2-s + (1.42 − 0.825i)3-s − 0.695·4-s + (−0.900 + 0.519i)5-s + (−1.07 − 1.86i)6-s + (−0.204 − 0.117i)7-s − 0.396i·8-s + (0.861 − 1.49i)9-s + (0.676 + 1.17i)10-s − 1.20·11-s + (−0.994 + 0.574i)12-s + (−0.437 + 0.757i)13-s + (−0.153 + 0.265i)14-s + (−0.857 + 1.48i)15-s − 1.21·16-s + (0.356 − 0.618i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.746708 + 1.04791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.746708 + 1.04791i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.42e8 + 3.58e7i)T \) |
| good | 2 | \( 1 + 41.6iT - 1.02e3T^{2} \) |
| 3 | \( 1 + (-347. + 200. i)T + (2.95e4 - 5.11e4i)T^{2} \) |
| 5 | \( 1 + (2.81e3 - 1.62e3i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 7 | \( 1 + (3.43e3 + 1.98e3i)T + (1.41e8 + 2.44e8i)T^{2} \) |
| 11 | \( 1 + 1.94e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + (1.62e5 - 2.81e5i)T + (-6.89e10 - 1.19e11i)T^{2} \) |
| 17 | \( 1 + (-5.06e5 + 8.77e5i)T + (-1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (9.44e5 - 5.45e5i)T + (3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (3.26e6 + 5.65e6i)T + (-2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 + (5.32e6 + 3.07e6i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (8.01e6 + 1.38e7i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + (1.45e7 - 8.37e6i)T + (2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 - 4.23e7T + 1.34e16T^{2} \) |
| 47 | \( 1 - 4.42e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + (2.99e8 + 5.17e8i)T + (-8.74e16 + 1.51e17i)T^{2} \) |
| 59 | \( 1 + 5.57e8T + 5.11e17T^{2} \) |
| 61 | \( 1 + (-1.14e9 - 6.60e8i)T + (3.56e17 + 6.17e17i)T^{2} \) |
| 67 | \( 1 + (7.23e8 + 1.25e9i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + (1.02e9 + 5.91e8i)T + (1.62e18 + 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-3.09e8 - 1.78e8i)T + (2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (-2.08e9 + 3.61e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-6.55e8 - 1.13e9i)T + (-7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 + (-3.01e9 + 1.74e9i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 + 1.56e10T + 7.37e19T^{2} \) |
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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
−12.80837012142300389364632525025, −11.90600777369830722251200546873, −10.59653113128837040834734236146, −9.332667739423773982773060266524, −7.921855915033195580346439010001, −7.02459440702688460621961744201, −3.99279693447984575868442715157, −2.88324405186963896285145788844, −2.07000019906607298684040575179, −0.31723202268357552883605915807,
2.66707227509809931324178947071, 4.15082500273642471216464707832, 5.45128179423751875642349056657, 7.65055648574902455540628128960, 8.073061557519900002064304823610, 9.177833676010481264289519373338, 10.62095438614032212455053459239, 12.62087668964672935333537413832, 13.91211946290179182520362887653, 14.97512338738819796362129137756
