| L(s) = 1 | + 724. i·2-s − 5.24e5·4-s − 1.19e7i·5-s + 5.47e8·7-s − 3.79e8i·8-s + 8.62e9·10-s + 4.67e10i·11-s + 2.59e11·13-s + 3.96e11i·14-s + 2.74e11·16-s + 1.68e12i·17-s + 3.44e11·19-s + 6.24e12i·20-s − 3.38e13·22-s + 2.77e13i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 1.21i·5-s + 1.93·7-s − 0.353i·8-s + 0.862·10-s + 1.80i·11-s + 1.87·13-s + 1.37i·14-s + 0.250·16-s + 0.837i·17-s + 0.0561·19-s + 0.609i·20-s − 1.27·22-s + 0.668i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(3.041632182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.041632182\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 724. iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.19e7iT - 9.53e13T^{2} \) |
| 7 | \( 1 - 5.47e8T + 7.97e16T^{2} \) |
| 11 | \( 1 - 4.67e10iT - 6.72e20T^{2} \) |
| 13 | \( 1 - 2.59e11T + 1.90e22T^{2} \) |
| 17 | \( 1 - 1.68e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 - 3.44e11T + 3.75e25T^{2} \) |
| 23 | \( 1 - 2.77e13iT - 1.71e27T^{2} \) |
| 29 | \( 1 - 2.21e14iT - 1.76e29T^{2} \) |
| 31 | \( 1 - 2.59e14T + 6.71e29T^{2} \) |
| 37 | \( 1 + 3.51e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + 6.48e15iT - 1.80e32T^{2} \) |
| 43 | \( 1 + 2.03e16T + 4.67e32T^{2} \) |
| 47 | \( 1 - 1.46e16iT - 2.76e33T^{2} \) |
| 53 | \( 1 - 3.30e17iT - 3.05e34T^{2} \) |
| 59 | \( 1 - 3.00e17iT - 2.61e35T^{2} \) |
| 61 | \( 1 + 3.04e17T + 5.08e35T^{2} \) |
| 67 | \( 1 + 5.28e17T + 3.32e36T^{2} \) |
| 71 | \( 1 - 4.65e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 + 2.80e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + 3.26e18T + 8.96e37T^{2} \) |
| 83 | \( 1 + 1.58e19iT - 2.40e38T^{2} \) |
| 89 | \( 1 + 4.86e19iT - 9.72e38T^{2} \) |
| 97 | \( 1 + 9.93e19T + 5.43e39T^{2} \) |
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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
−11.79791780589495336179766290964, −10.51975975925036642478198707736, −8.932728432466414818538133710952, −8.334548768581401455892705295666, −7.34483132298823711513976993038, −5.71528860758421731170268086892, −4.76922079771129977992238582516, −4.11561292346016311517909590677, −1.58355375733648209243760159131, −1.31205968725517262049174930947,
0.64638232758776867664326291155, 1.61775988165737647534785105510, 2.88164895737497101743855938789, 3.81154973879763892724991315165, 5.23093947723189029219370423231, 6.42934406065727871109766996198, 8.068514185765141518238372837853, 8.672952385932572685048165602120, 10.53997258197317454419470905514, 11.20101981109478457712585564469
