| L(s) = 1 | + (24.3 − 11.7i)3-s + (76.0 + 43.8i)5-s + (−287. − 498. i)7-s + (452. − 571. i)9-s + (−759. + 438. i)11-s + (1.45e3 − 2.52e3i)13-s + (2.36e3 + 172. i)15-s + 1.89e3i·17-s + 1.61e3·19-s + (−1.28e4 − 8.73e3i)21-s + (−7.09e3 − 4.09e3i)23-s + (−3.95e3 − 6.85e3i)25-s + (4.27e3 − 1.92e4i)27-s + (3.91e4 − 2.26e4i)29-s + (−8.29e3 + 1.43e4i)31-s + ⋯ |
| L(s) = 1 | + (0.900 − 0.435i)3-s + (0.608 + 0.351i)5-s + (−0.839 − 1.45i)7-s + (0.620 − 0.784i)9-s + (−0.570 + 0.329i)11-s + (0.663 − 1.14i)13-s + (0.700 + 0.0511i)15-s + 0.384i·17-s + 0.234·19-s + (−1.38 − 0.943i)21-s + (−0.582 − 0.336i)23-s + (−0.253 − 0.438i)25-s + (0.217 − 0.976i)27-s + (1.60 − 0.927i)29-s + (−0.278 + 0.482i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0286 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0286 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.69676 - 1.64874i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69676 - 1.64874i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-24.3 + 11.7i)T \) |
| good | 5 | \( 1 + (-76.0 - 43.8i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (287. + 498. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (759. - 438. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-1.45e3 + 2.52e3i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 - 1.89e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.61e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (7.09e3 + 4.09e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-3.91e4 + 2.26e4i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (8.29e3 - 1.43e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 - 3.73e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (8.25e3 + 4.76e3i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (6.88e4 + 1.19e5i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-4.80e3 + 2.77e3i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 - 2.66e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-2.15e5 - 1.24e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.82e5 - 3.15e5i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.30e5 - 2.26e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 1.31e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.59e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-4.43e5 - 7.68e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-4.92e5 + 2.84e5i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 - 8.37e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (4.30e5 + 7.44e5i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−13.48717692531583659381070648987, −12.44552650306328343326541554939, −10.35815153684299625223461899735, −10.05114586359447894839614895682, −8.361123783946428652506388959696, −7.27271348524318552324432145072, −6.17819701643648529390554148980, −3.94824418954812706577183587065, −2.67673028380666646179432581283, −0.832579057472065295162542016253,
2.00437508005944500682166918511, 3.25984553340654912317989523099, 5.08838285653865932517304360342, 6.39974276496802442128671067834, 8.280132180527018732553356853066, 9.197003443544671225414290427612, 9.857197574838109463731419606101, 11.51703652067402772704606267868, 12.86446841771674148784116383413, 13.64656197349903645718213285700
