| L(s) = 1 | − 10.4i·2-s − 477. i·3-s + 914.·4-s − 5.02e3i·5-s − 4.98e3·6-s + 7.30e3i·7-s − 2.02e4i·8-s − 1.68e5·9-s − 5.25e4·10-s − 1.41e5·11-s − 4.36e5i·12-s + 5.66e5·13-s + 7.63e4·14-s − 2.39e6·15-s + 7.25e5·16-s − 3.70e5·17-s + ⋯ |
| L(s) = 1 | − 0.326i·2-s − 1.96i·3-s + 0.893·4-s − 1.60i·5-s − 0.641·6-s + 0.434i·7-s − 0.618i·8-s − 2.86·9-s − 0.525·10-s − 0.879·11-s − 1.75i·12-s + 1.52·13-s + 0.141·14-s − 3.16·15-s + 0.691·16-s − 0.260·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.846337 + 1.96728i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.846337 + 1.96728i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.01e8 - 1.06e8i)T \) |
| good | 2 | \( 1 + 10.4iT - 1.02e3T^{2} \) |
| 3 | \( 1 + 477. iT - 5.90e4T^{2} \) |
| 5 | \( 1 + 5.02e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 - 7.30e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 1.41e5T + 2.59e10T^{2} \) |
| 13 | \( 1 - 5.66e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 3.70e5T + 2.01e12T^{2} \) |
| 19 | \( 1 + 1.53e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 9.51e6T + 4.14e13T^{2} \) |
| 29 | \( 1 + 1.65e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 2.42e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 1.62e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 + 1.33e8T + 1.34e16T^{2} \) |
| 47 | \( 1 + 2.40e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + 2.67e7T + 1.74e17T^{2} \) |
| 59 | \( 1 - 6.95e8T + 5.11e17T^{2} \) |
| 61 | \( 1 - 9.17e8iT - 7.13e17T^{2} \) |
| 67 | \( 1 - 1.19e9T + 1.82e18T^{2} \) |
| 71 | \( 1 + 4.68e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 2.00e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 1.34e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 1.99e9T + 1.55e19T^{2} \) |
| 89 | \( 1 + 2.98e8iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 1.69e9T + 7.37e19T^{2} \) |
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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
−13.07237157566193644020989069411, −12.02609350583114829942981175855, −11.19376086108478826804577521130, −8.794377885163670396790949430471, −7.994590792653838710191693826042, −6.60058083939041294173542403711, −5.47758059280382083885069793298, −2.72142403594638062032229266308, −1.46365608167024137391289069950, −0.69888453517419502166610947746,
2.80662577807578711361791961341, 3.63511196854535479978442920713, 5.48571482195289755437959587009, 6.71834211282800145878868079703, 8.370109871973042794186848249850, 10.14007550097415224341931463456, 10.83339138966256101113879896590, 11.25528027133252914020555848318, 13.93930476017835956106329477954, 14.91111638405226129497112726631
