| L(s) = 1 | + (5.69 + 3.28i)2-s + (−24.6 − 11.0i)3-s + (−10.3 − 17.9i)4-s + (−57.4 − 33.1i)5-s + (−104. − 143. i)6-s + (−329. − 93.7i)7-s − 557. i·8-s + (486. + 542. i)9-s + (−218. − 377. i)10-s + (603. − 348. i)11-s + (57.8 + 556. i)12-s − 824.·13-s + (−1.57e3 − 1.61e3i)14-s + (1.05e3 + 1.45e3i)15-s + (1.16e3 − 2.02e3i)16-s + (−4.78e3 + 2.76e3i)17-s + ⋯ |
| L(s) = 1 | + (0.712 + 0.411i)2-s + (−0.913 − 0.407i)3-s + (−0.161 − 0.280i)4-s + (−0.459 − 0.265i)5-s + (−0.482 − 0.665i)6-s + (−0.961 − 0.273i)7-s − 1.08i·8-s + (0.667 + 0.744i)9-s + (−0.218 − 0.377i)10-s + (0.453 − 0.261i)11-s + (0.0334 + 0.322i)12-s − 0.375·13-s + (−0.572 − 0.590i)14-s + (0.311 + 0.429i)15-s + (0.285 − 0.494i)16-s + (−0.973 + 0.562i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.360283 - 0.711536i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.360283 - 0.711536i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (24.6 + 11.0i)T \) |
| 7 | \( 1 + (329. + 93.7i)T \) |
| good | 2 | \( 1 + (-5.69 - 3.28i)T + (32 + 55.4i)T^{2} \) |
| 5 | \( 1 + (57.4 + 33.1i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-603. + 348. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 824.T + 4.82e6T^{2} \) |
| 17 | \( 1 + (4.78e3 - 2.76e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-2.52e3 + 4.37e3i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-1.72e4 - 9.94e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 2.34e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + (2.31e4 + 4.01e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-2.41e4 + 4.18e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 - 1.59e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 2.17e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (3.30e4 + 1.90e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (1.93e5 - 1.11e5i)T + (1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-2.71e5 + 1.56e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.26e5 - 2.19e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-8.29e4 - 1.43e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 2.98e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + (2.21e3 + 3.83e3i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-3.10e5 + 5.37e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 4.14e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (6.49e5 + 3.75e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 2.12e5T + 8.32e11T^{2} \) |
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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
−16.23702654897151201410786026612, −15.26026380592709177521192765236, −13.50089733362094657326720908697, −12.74674104464440604190864762246, −11.22673248698736058118019308714, −9.567474751049069041737047043448, −7.11194347335385051146059087650, −5.91512786216121192714814976239, −4.27816780305132899162080806737, −0.45012510375427407930284705194,
3.39506325130735873317482420977, 4.96695806176595980436556141288, 6.85028541178493549585081807202, 9.225282785317760993849902050171, 10.95266147236042108568138032797, 12.09031895925343483892897246733, 12.96933718439553821126730342488, 14.71713581240765987880030858576, 16.04550798278933531656499991062, 17.13714171686770225205598003513
