| L(s) = 1 | + (5.65 − 9.79i)2-s + (−40.5 + 23.3i)3-s + (−63.9 − 110. i)4-s + (−127. − 73.6i)5-s + 529. i·6-s + (1.14e3 + 2.10e3i)7-s − 1.44e3·8-s + (1.09e3 − 1.89e3i)9-s + (−1.44e3 + 833. i)10-s + (6.28e3 + 1.08e4i)11-s + (5.18e3 + 2.99e3i)12-s − 4.58e4i·13-s + (2.71e4 + 679. i)14-s + 6.88e3·15-s + (−8.19e3 + 1.41e4i)16-s + (6.39e4 − 3.69e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.5 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.204 − 0.117i)5-s + 0.408i·6-s + (0.478 + 0.878i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.144 + 0.0833i)10-s + (0.429 + 0.744i)11-s + (0.249 + 0.144i)12-s − 1.60i·13-s + (0.706 + 0.0176i)14-s + 0.136·15-s + (−0.125 + 0.216i)16-s + (0.766 − 0.442i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.84039 - 0.646212i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84039 - 0.646212i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-5.65 + 9.79i)T \) |
| 3 | \( 1 + (40.5 - 23.3i)T \) |
| 7 | \( 1 + (-1.14e3 - 2.10e3i)T \) |
| good | 5 | \( 1 + (127. + 73.6i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-6.28e3 - 1.08e4i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 4.58e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-6.39e4 + 3.69e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-2.18e5 - 1.26e5i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-1.62e5 + 2.81e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 1.07e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + (5.80e5 - 3.35e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-7.92e5 + 1.37e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 - 4.20e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.44e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-3.10e6 - 1.79e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-1.26e6 - 2.19e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (2.04e7 - 1.18e7i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (2.48e6 + 1.43e6i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (9.23e5 + 1.59e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 2.54e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.76e7 + 1.02e7i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-3.52e7 + 6.11e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 1.20e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-2.83e7 - 1.63e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + 8.92e7iT - 7.83e15T^{2} \) |
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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
−14.21095015557594510876860832466, −12.41689450641159501925534290092, −12.04358628191799474439919737890, −10.61518114512134049610694975692, −9.530590917732401864093688745468, −7.901155846171855309472817292358, −5.82433398738662051221616277719, −4.76784747070411731985092899088, −2.97754206484919435575482504436, −0.999415734074814642250205141015,
1.09481274799132590758961875956, 3.72160319773981924198411110799, 5.19340105298656708228585635238, 6.73450292587883615742714958796, 7.66689718013875155486388105268, 9.322290910166045823253789685318, 11.14150371377149413476649690711, 11.91488344237529653438860283057, 13.63824326897543925707605732534, 14.10340541437475609793544848223
