| L(s) = 1 | + 0.461i·2-s + (22.0 + 15.5i)3-s + 63.7·4-s + 70.0i·5-s + (−7.19 + 10.1i)6-s + 296.·7-s + 58.9i·8-s + (242. + 687. i)9-s − 32.3·10-s − 1.24e3i·11-s + (1.40e3 + 994. i)12-s − 2.89e3·13-s + 136. i·14-s + (−1.09e3 + 1.54e3i)15-s + 4.05e3·16-s + 1.19e3i·17-s + ⋯ |
| L(s) = 1 | + 0.0576i·2-s + (0.816 + 0.577i)3-s + 0.996·4-s + 0.560i·5-s + (−0.0333 + 0.0470i)6-s + 0.864·7-s + 0.115i·8-s + (0.332 + 0.942i)9-s − 0.0323·10-s − 0.937i·11-s + (0.813 + 0.575i)12-s − 1.31·13-s + 0.0498i·14-s + (−0.323 + 0.457i)15-s + 0.990·16-s + 0.242i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.64348 + 1.36795i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.64348 + 1.36795i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-22.0 - 15.5i)T \) |
| 17 | \( 1 - 1.19e3iT \) |
| good | 2 | \( 1 - 0.461iT - 64T^{2} \) |
| 5 | \( 1 - 70.0iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 296.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.24e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.89e3T + 4.82e6T^{2} \) |
| 19 | \( 1 - 3.45e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 2.09e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 4.38e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 1.56e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 5.20e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 1.35e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.91e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.38e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.59e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 2.61e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.15e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.47e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 8.44e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.29e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 8.37e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 4.85e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 2.78e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 2.31e5T + 8.32e11T^{2} \) |
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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
−14.67782434635763190526512310382, −13.64461224306223173115901814067, −11.79635836172090247396154042713, −10.90120763815226268871719158060, −9.775410742943872250245343119668, −8.148471115990954300941624667835, −7.21384085516388074033186526143, −5.34523826021561221711233637108, −3.38119574331882075064578193994, −2.05726524105390905381281587475,
1.42426031927839091925402624190, 2.65696729437365178757575479409, 4.83830615884208094933173334200, 6.90571620242803621333662865370, 7.73941540891718132872842799240, 9.090043118662467697412508970653, 10.55353553443497918496838900485, 12.16951625913858082315986246723, 12.57348602610952265396605405623, 14.40498488973305892497847252323
