| L(s) = 1 | + (−14.5 − 22.7i)3-s + (26.1 − 15.1i)5-s + (301. − 522. i)7-s + (−307. + 661. i)9-s + (1.12e3 + 652. i)11-s + (−1.08e3 − 1.88e3i)13-s + (−724. − 376. i)15-s − 4.92e3i·17-s − 3.40e3·19-s + (−1.62e4 + 718. i)21-s + (−1.14e4 + 6.60e3i)23-s + (−7.35e3 + 1.27e4i)25-s + (1.95e4 − 2.60e3i)27-s + (−3.18e4 − 1.83e4i)29-s + (−1.77e4 − 3.07e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.537 − 0.843i)3-s + (0.209 − 0.120i)5-s + (0.878 − 1.52i)7-s + (−0.421 + 0.906i)9-s + (0.848 + 0.489i)11-s + (−0.495 − 0.858i)13-s + (−0.214 − 0.111i)15-s − 1.00i·17-s − 0.496·19-s + (−1.75 + 0.0775i)21-s + (−0.940 + 0.542i)23-s + (−0.470 + 0.815i)25-s + (0.991 − 0.132i)27-s + (−1.30 − 0.753i)29-s + (−0.596 − 1.03i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.423i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.905 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.270737 - 1.21817i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.270737 - 1.21817i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (14.5 + 22.7i)T \) |
| good | 5 | \( 1 + (-26.1 + 15.1i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-301. + 522. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.12e3 - 652. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (1.08e3 + 1.88e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + 4.92e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 3.40e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (1.14e4 - 6.60e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (3.18e4 + 1.83e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.77e4 + 3.07e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 8.57e3T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-6.74e4 + 3.89e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (3.14e4 - 5.45e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-1.00e5 - 5.80e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 4.84e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-1.76e5 + 1.01e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.39e4 - 2.41e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-2.33e5 - 4.04e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 3.28e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.06e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-2.83e5 + 4.91e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-5.56e5 - 3.21e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + 2.20e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-1.14e5 + 1.98e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−13.05179579629340436538749168304, −11.76170810785849407916231501632, −10.93228649327890607297110949775, −9.664601768186538680865177333818, −7.76961816873446530319722597718, −7.23190339248193381784204613280, −5.64025464881605230092568055135, −4.21915743663517320064311420507, −1.82235595361751675189067359814, −0.51187355779395558232677013577,
2.01000065078153254952807184752, 4.02188651202953665969679485819, 5.40112350394059964085266662465, 6.36955872028189927398387471427, 8.510632961274186548467774291367, 9.262733198628075584733085651977, 10.65231157632774210143965346412, 11.69323667742984002225003136834, 12.35741813145996817158508119998, 14.40082920810293226838313607440
