| L(s) = 1 | − 724. i·2-s − 5.24e5·4-s − 1.04e7i·5-s + 3.13e8·7-s + 3.79e8i·8-s − 7.54e9·10-s − 5.11e9i·11-s − 8.38e10·13-s − 2.27e11i·14-s + 2.74e11·16-s − 3.93e11i·17-s + 2.35e12·19-s + 5.46e12i·20-s − 3.70e12·22-s + 3.96e13i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 1.06i·5-s + 1.11·7-s + 0.353i·8-s − 0.754·10-s − 0.197i·11-s − 0.608·13-s − 0.785i·14-s + 0.250·16-s − 0.194i·17-s + 0.384·19-s + 0.533i·20-s − 0.139·22-s + 0.956i·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
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(0.04816427129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04816427129\) |
| \(L(11)\) |
|
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 | \( 1 + 1.04e7iT - 9.53e13T^{2} \) |
| 7 | \( 1 - 3.13e8T + 7.97e16T^{2} \) |
| 11 | \( 1 + 5.11e9iT - 6.72e20T^{2} \) |
| 13 | \( 1 + 8.38e10T + 1.90e22T^{2} \) |
| 17 | \( 1 + 3.93e11iT - 4.06e24T^{2} \) |
| 19 | \( 1 - 2.35e12T + 3.75e25T^{2} \) |
| 23 | \( 1 - 3.96e13iT - 1.71e27T^{2} \) |
| 29 | \( 1 - 2.25e14iT - 1.76e29T^{2} \) |
| 31 | \( 1 + 7.57e14T + 6.71e29T^{2} \) |
| 37 | \( 1 + 7.93e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + 1.78e16iT - 1.80e32T^{2} \) |
| 43 | \( 1 + 1.01e16T + 4.67e32T^{2} \) |
| 47 | \( 1 - 2.93e16iT - 2.76e33T^{2} \) |
| 53 | \( 1 + 6.24e15iT - 3.05e34T^{2} \) |
| 59 | \( 1 + 7.16e17iT - 2.61e35T^{2} \) |
| 61 | \( 1 + 1.56e17T + 5.08e35T^{2} \) |
| 67 | \( 1 + 7.08e17T + 3.32e36T^{2} \) |
| 71 | \( 1 + 2.46e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 + 7.33e18T + 1.84e37T^{2} \) |
| 79 | \( 1 - 5.21e18T + 8.96e37T^{2} \) |
| 83 | \( 1 - 9.97e18iT - 2.40e38T^{2} \) |
| 89 | \( 1 + 2.83e19iT - 9.72e38T^{2} \) |
| 97 | \( 1 + 6.43e18T + 5.43e39T^{2} \) |
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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
−10.70126730273471253011091881324, −9.367296097952459882053063042638, −8.529574315547329072048117768034, −7.41676657107653679794558315055, −5.36893428444880731568726217218, −4.80897075017023748627387762069, −3.49805296225939882059976158443, −1.94617895031329223007232146883, −1.19078223595578060724826391179, −0.009492905021890612366583974061,
1.61356496761149141132400301042, 2.89566306416518679010386375628, 4.32926077700298707820946706437, 5.40154787991503716086769652457, 6.71704362424980714241936894973, 7.54038777927295022999657716736, 8.598772433259864407534671910346, 10.03201440400787169766284247660, 11.00431168008697170160031069997, 12.15976435630757274902995064971
