| L(s) = 1 | − 7.09i·3-s − 2.59·7-s + 30.6·9-s + 7.95·11-s − 54.6·13-s + 28.7i·17-s − 316.·19-s + 18.3i·21-s + 846.·23-s − 792. i·27-s + 766. i·29-s − 1.19e3i·31-s − 56.4i·33-s − 2.43e3·37-s + 387. i·39-s + ⋯ |
| L(s) = 1 | − 0.788i·3-s − 0.0528·7-s + 0.377·9-s + 0.0657·11-s − 0.323·13-s + 0.0995i·17-s − 0.877·19-s + 0.0417i·21-s + 1.60·23-s − 1.08i·27-s + 0.911i·29-s − 1.24i·31-s − 0.0518i·33-s − 1.77·37-s + 0.255i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.607460713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.607460713\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 7.09iT - 81T^{2} \) |
| 7 | \( 1 + 2.59T + 2.40e3T^{2} \) |
| 11 | \( 1 - 7.95T + 1.46e4T^{2} \) |
| 13 | \( 1 + 54.6T + 2.85e4T^{2} \) |
| 17 | \( 1 - 28.7iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 316.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 846.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 766. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.19e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 2.43e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.43e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.33e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.19e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 3.04e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 455.T + 1.21e7T^{2} \) |
| 61 | \( 1 - 3.92e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 4.82e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 874. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 7.68e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 1.47e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 6.84e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.31e4T + 6.27e7T^{2} \) |
| 97 | \( 1 - 8.60e3iT - 8.85e7T^{2} \) |
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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
−9.248030542842477559148453613467, −8.533066802558389513877072537429, −7.41035476256194823091553213319, −6.98009859892635282294553590092, −5.99752428890600522838191680446, −4.91701170020515475683888163234, −3.86180600085002960354515015943, −2.56072042166678735985661021499, −1.54523078427723293385305670668, −0.39038261935295367369265931664,
1.16741015853439745720503248061, 2.59176451860281281571697753741, 3.72031245515733009994228526741, 4.60230225093739911855845234045, 5.36778439548858794857056743187, 6.61658368272782366164825622900, 7.33376490005029591051742409363, 8.518966080863109593382840070589, 9.224243557780943530330925319441, 10.03093108620269586416554418019
