| L(s) = 1 | + (15.4 + 1.94i)3-s − 101.·5-s + 200. i·7-s + (235. + 60.0i)9-s + 350. i·11-s + 744. i·13-s + (−1.56e3 − 196. i)15-s − 1.44e3i·17-s − 1.31e3·19-s + (−389. + 3.09e3i)21-s − 2.04e3·23-s + 7.09e3·25-s + (3.52e3 + 1.38e3i)27-s + 3.78e3·29-s − 1.42e3i·31-s + ⋯ |
| L(s) = 1 | + (0.992 + 0.124i)3-s − 1.80·5-s + 1.54i·7-s + (0.968 + 0.247i)9-s + 0.872i·11-s + 1.22i·13-s + (−1.79 − 0.225i)15-s − 1.21i·17-s − 0.834·19-s + (−0.192 + 1.53i)21-s − 0.804·23-s + 2.27·25-s + (0.930 + 0.366i)27-s + 0.835·29-s − 0.266i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 + 0.613i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4971740096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4971740096\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-15.4 - 1.94i)T \) |
| good | 5 | \( 1 + 101.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 200. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 350. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 744. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.44e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.42e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 2.61e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.08e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.61e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.41e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 539.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.37e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 5.04e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.32e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 7.58e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 9.73e3iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.15e5T + 8.58e9T^{2} \) |
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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
−11.33992618012308402714868983612, −9.931326666940114410460133384092, −8.949377949800578146826759433155, −8.448935727214192415681026690588, −7.50671338648028454146232031215, −6.67483521314644697559908166009, −4.83708399489500471505901212429, −4.15482169035510641745196155151, −2.96504559318844753402171414721, −1.96425447708763042090107118232,
0.11838964934203209428105228704, 1.13164499588923528579915939216, 3.14627013737502745954898866533, 3.78422998235705049022287943543, 4.49410668002317248675228014407, 6.47339880994914827936171700199, 7.49361568195150353102812377353, 8.123515316356021487114938885446, 8.527783311378026022882882532118, 10.28239740300238924028973457940
