| L(s) = 1 | − 32.4i·3-s − 199.·5-s − 19.6i·7-s − 326.·9-s + 924. i·11-s + 1.55e3·13-s + 6.46e3i·15-s + 5.14e3·17-s − 1.69e3i·19-s − 639.·21-s + 1.92e4i·23-s + 2.40e4·25-s − 1.30e4i·27-s − 1.65e4·29-s − 7.55e3i·31-s + ⋯ |
| L(s) = 1 | − 1.20i·3-s − 1.59·5-s − 0.0573i·7-s − 0.448·9-s + 0.694i·11-s + 0.705·13-s + 1.91i·15-s + 1.04·17-s − 0.247i·19-s − 0.0690·21-s + 1.57i·23-s + 1.53·25-s − 0.663i·27-s − 0.680·29-s − 0.253i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.381359777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.381359777\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 32.4iT - 729T^{2} \) |
| 5 | \( 1 + 199.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 19.6iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 924. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.55e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 5.14e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 1.69e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.92e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.65e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 7.55e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 2.89e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 5.21e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 5.89e3iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 6.44e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.97e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.42e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 9.64e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 7.52e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 5.56e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 2.85e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 3.42e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 9.29e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 4.34e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 6.43e5T + 8.32e11T^{2} \) |
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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
−11.13115544772790645772999822805, −9.712318050648012990959290924101, −8.384367163094486725766336752666, −7.53851243448910115064129753602, −7.20098081859718490340927064413, −5.81907481648215846147062675620, −4.30086537168047509179908294079, −3.27313808261264690460811396347, −1.62348878504049525785746871340, −0.54122833473784017410696098394,
0.792564053899169156112063696314, 3.14447242309211294834742953006, 3.89499015667208496323507373916, 4.70824956424893741947504244666, 6.03767813373085183253370620217, 7.51379837113118652723812809609, 8.354049996647758190156765084847, 9.227188375573746698817553280443, 10.49088134477781166663108689481, 11.01619681506519122867034988433
