| L(s) = 1 | + (−4.89 − 2.82i)2-s + (−22.5 + 14.8i)3-s + (15.9 + 27.7i)4-s + (156. − 90.6i)5-s + (152. − 9.01i)6-s + (104. − 180. i)7-s − 181. i·8-s + (287. − 669. i)9-s − 1.02e3·10-s + (2.30e3 + 1.32e3i)11-s + (−772. − 387. i)12-s + (−438. − 759. i)13-s + (−1.02e3 + 590. i)14-s + (−2.19e3 + 4.37e3i)15-s + (−512. + 886. i)16-s − 4.42e3i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.834 + 0.550i)3-s + (0.249 + 0.433i)4-s + (1.25 − 0.724i)5-s + (0.705 − 0.0417i)6-s + (0.304 − 0.526i)7-s − 0.353i·8-s + (0.394 − 0.918i)9-s − 1.02·10-s + (1.72 + 0.998i)11-s + (−0.447 − 0.223i)12-s + (−0.199 − 0.345i)13-s + (−0.372 + 0.215i)14-s + (−0.649 + 1.29i)15-s + (−0.125 + 0.216i)16-s − 0.901i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.06095 - 0.316580i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06095 - 0.316580i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 + 2.82i)T \) |
| 3 | \( 1 + (22.5 - 14.8i)T \) |
| good | 5 | \( 1 + (-156. + 90.6i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-104. + 180. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-2.30e3 - 1.32e3i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (438. + 759. i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + 4.42e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 4.19e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.03e4 + 6.00e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.28e3 - 1.32e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (2.90e3 + 5.02e3i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 4.15e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (6.47e4 - 3.73e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (7.30e4 - 1.26e5i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (2.23e4 + 1.28e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + 1.97e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (3.13e4 - 1.80e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.19e4 - 2.07e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.76e5 + 3.05e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 4.96e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.82e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (1.93e5 - 3.34e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-3.59e5 - 2.07e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + 4.05e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (1.78e5 - 3.08e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−17.17871231696404850457973555412, −16.66348232388976480140557816013, −14.69779421784529223433934039204, −12.89116441520318149953780794593, −11.56735318939454563878435593924, −10.01713059621659912916558875343, −9.205319456922719782726839582936, −6.62502861487833775361075059611, −4.66882085037931787228778064770, −1.24705781352702887841057263612,
1.65792738516953793636586036853, 5.84535736133848716691370604527, 6.72073395963298167458554111107, 8.916765388414434587464367624836, 10.53048758410218736014418233441, 11.75393937354533707458213987525, 13.64117834681327302449944878316, 14.82352001146554451657898141781, 16.85070456645013304992978155377, 17.34046475468800231707097307462
