| L(s) = 1 | + (−15.9 − 1.12i)2-s + (−33.0 + 33.0i)3-s + (253. + 35.9i)4-s + (156. − 156. i)5-s + (565. − 490. i)6-s − 2.21e3·7-s + (−4.00e3 − 859. i)8-s − 2.18e3i·9-s + (−2.67e3 + 2.32e3i)10-s + (1.76e4 + 1.76e4i)11-s + (−9.57e3 + 7.19e3i)12-s + (1.51e4 + 1.51e4i)13-s + (3.53e4 + 2.49e3i)14-s + 1.03e4i·15-s + (6.29e4 + 1.82e4i)16-s − 1.48e5·17-s + ⋯ |
| L(s) = 1 | + (−0.997 − 0.0704i)2-s + (−0.408 + 0.408i)3-s + (0.990 + 0.140i)4-s + (0.250 − 0.250i)5-s + (0.435 − 0.378i)6-s − 0.922·7-s + (−0.977 − 0.209i)8-s − 0.333i·9-s + (−0.267 + 0.232i)10-s + (1.20 + 1.20i)11-s + (−0.461 + 0.346i)12-s + (0.530 + 0.530i)13-s + (0.919 + 0.0649i)14-s + 0.204i·15-s + (0.960 + 0.278i)16-s − 1.77·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.102152 - 0.212527i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.102152 - 0.212527i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (15.9 + 1.12i)T \) |
| 3 | \( 1 + (33.0 - 33.0i)T \) |
| good | 5 | \( 1 + (-156. + 156. i)T - 3.90e5iT^{2} \) |
| 7 | \( 1 + 2.21e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + (-1.76e4 - 1.76e4i)T + 2.14e8iT^{2} \) |
| 13 | \( 1 + (-1.51e4 - 1.51e4i)T + 8.15e8iT^{2} \) |
| 17 | \( 1 + 1.48e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + (-1.03e5 + 1.03e5i)T - 1.69e10iT^{2} \) |
| 23 | \( 1 + 1.98e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (8.31e5 + 8.31e5i)T + 5.00e11iT^{2} \) |
| 31 | \( 1 + 1.08e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-7.56e5 + 7.56e5i)T - 3.51e12iT^{2} \) |
| 41 | \( 1 + 5.35e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + (7.52e5 + 7.52e5i)T + 1.16e13iT^{2} \) |
| 47 | \( 1 + 2.04e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + (8.95e6 - 8.95e6i)T - 6.22e13iT^{2} \) |
| 59 | \( 1 + (1.61e6 + 1.61e6i)T + 1.46e14iT^{2} \) |
| 61 | \( 1 + (5.64e6 + 5.64e6i)T + 1.91e14iT^{2} \) |
| 67 | \( 1 + (-5.04e6 + 5.04e6i)T - 4.06e14iT^{2} \) |
| 71 | \( 1 + 2.68e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 5.97e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 5.23e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (-2.72e7 + 2.72e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 - 4.38e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.81e7T + 7.83e15T^{2} \) |
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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
−13.32929648199370769443121951432, −11.98879431234135876847151777859, −11.04031775944135945699395287717, −9.424406379261210426231327960003, −9.296603344393993029576625430652, −7.13895526590967278206423579371, −6.15506524443617106273530405464, −4.02335090420744329412188589722, −1.94302091190360372917972785014, −0.12901826950466273133118389832,
1.34873375443576229054864494007, 3.22306629964913571481121200697, 6.07252886916760846638594562096, 6.66278971014266135194195145953, 8.338264382236466785573870277150, 9.474241399478724552612647817568, 10.76844345611781977170453084084, 11.69113329405433611069348151463, 13.06096960408069248162220604228, 14.38093961862893991315396927454
