| L(s) = 1 | + (−2.73 − 4.73i)2-s + (−58.8 − 33.9i)3-s + (113. − 195. i)4-s + (−586. + 338. i)5-s + 372. i·6-s − 2.63e3·8-s + (−968. − 1.67e3i)9-s + (3.20e3 + 1.85e3i)10-s + (−7.99e3 + 1.38e4i)11-s + (−1.33e4 + 7.68e3i)12-s + 7.14e3i·13-s + 4.60e4·15-s + (−2.17e4 − 3.76e4i)16-s + (−3.76e4 − 2.17e4i)17-s + (−5.30e3 + 9.18e3i)18-s + (1.66e5 − 9.61e4i)19-s + ⋯ |
| L(s) = 1 | + (−0.170 − 0.296i)2-s + (−0.726 − 0.419i)3-s + (0.441 − 0.764i)4-s + (−0.938 + 0.541i)5-s + 0.287i·6-s − 0.643·8-s + (−0.147 − 0.255i)9-s + (0.320 + 0.185i)10-s + (−0.545 + 0.945i)11-s + (−0.641 + 0.370i)12-s + 0.249i·13-s + 0.909·15-s + (−0.331 − 0.574i)16-s + (−0.451 − 0.260i)17-s + (−0.0505 + 0.0874i)18-s + (1.27 − 0.737i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.673544 + 0.157785i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.673544 + 0.157785i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (2.73 + 4.73i)T + (-128 + 221. i)T^{2} \) |
| 3 | \( 1 + (58.8 + 33.9i)T + (3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (586. - 338. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (7.99e3 - 1.38e4i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 7.14e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (3.76e4 + 2.17e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.66e5 + 9.61e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-9.19e4 - 1.59e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 1.23e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-9.28e5 - 5.35e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (1.10e6 + 1.91e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 1.82e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 3.38e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (2.72e6 - 1.57e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (2.66e6 - 4.61e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.22e7 + 7.07e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (1.39e7 - 8.02e6i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.45e6 + 2.51e6i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 1.93e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.68e7 - 9.75e6i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-2.39e7 - 4.15e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 8.03e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-2.83e7 + 1.63e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 6.94e7iT - 7.83e15T^{2} \) |
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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
−14.05337828192232055663984782196, −12.30003056395378045794186781966, −11.57752513308801314400992279099, −10.72883219126117959863969028848, −9.390945231375709949755632295416, −7.41342764377450644952589151585, −6.53139066924745786013031473278, −4.99928279701079269850575357107, −2.86859491371959883756049010575, −0.996185859362118133434206075254,
0.36554779645242354319435582737, 3.08921151175231805525839058123, 4.65529482496998410253433185904, 6.13075188850461190539247298081, 7.81945871297945628508739526200, 8.513898996282356063446640635513, 10.47904464048932461310160733715, 11.58793703053153856299183464489, 12.25364422178976694818331538339, 13.70244576022571264306994402968
