| L(s) = 1 | + (26.9 − 0.680i)3-s + (158. + 91.5i)5-s + (99.1 + 171. i)7-s + (728. − 36.7i)9-s + (−759. + 438. i)11-s + (−1.07e3 + 1.86e3i)13-s + (4.34e3 + 2.36e3i)15-s − 5.77e3i·17-s − 2.65e3·19-s + (2.79e3 + 4.56e3i)21-s + (8.15e3 + 4.71e3i)23-s + (8.95e3 + 1.55e4i)25-s + (1.96e4 − 1.48e3i)27-s + (−8.03e3 + 4.64e3i)29-s + (3.18e3 − 5.51e3i)31-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.0251i)3-s + (1.26 + 0.732i)5-s + (0.288 + 0.500i)7-s + (0.998 − 0.0503i)9-s + (−0.570 + 0.329i)11-s + (−0.488 + 0.846i)13-s + (1.28 + 0.700i)15-s − 1.17i·17-s − 0.386·19-s + (0.301 + 0.493i)21-s + (0.670 + 0.387i)23-s + (0.573 + 0.992i)25-s + (0.997 − 0.0755i)27-s + (−0.329 + 0.190i)29-s + (0.106 − 0.185i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.97734 + 1.16941i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.97734 + 1.16941i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-26.9 + 0.680i)T \) |
| good | 5 | \( 1 + (-158. - 91.5i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-99.1 - 171. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (759. - 438. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (1.07e3 - 1.86e3i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + 5.77e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 2.65e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-8.15e3 - 4.71e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (8.03e3 - 4.64e3i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-3.18e3 + 5.51e3i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 - 9.74e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-9.26e4 - 5.35e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (5.09e4 + 8.82e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.45e5 - 8.41e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + 1.16e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (3.07e5 + 1.77e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.12e5 + 1.95e5i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.04e5 + 1.80e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 5.04e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 8.37e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-3.14e4 - 5.44e4i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (4.21e5 - 2.43e5i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 - 1.22e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (5.98e5 + 1.03e6i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−13.74389923791369650796842756229, −12.77970968186142438419107057278, −11.19680412914492039163832979775, −9.793953647700926582645524279351, −9.264988598562382673979429458945, −7.68910278027729681369585082823, −6.50952150869722061822264257438, −4.88461484094869698927944463592, −2.81897825789521451760267077439, −1.94309918526615930318456854083,
1.25013954568333347287395488592, 2.65574383597184421794606251732, 4.50318113722581460324168403100, 5.92678884605925990479772093009, 7.68743920566054975707495080501, 8.697753209288434787690170002180, 9.806526414064992814343133462784, 10.67073490372708750180884056469, 12.87062245300988186461142241524, 13.12109990000379515100324428810
