L(s) = 1 | + 17i·2-s − 81i·3-s − 33·4-s + 1.37e3·6-s + 3.79e3i·8-s − 6.56e3·9-s + 2.67e3i·12-s − 7.28e4·16-s − 2.11e4i·17-s − 1.11e5i·18-s + 2.03e5·19-s + 5.50e5i·23-s + 3.07e5·24-s + 5.31e5i·27-s + 1.83e6·31-s − 2.68e5i·32-s + ⋯ |
L(s) = 1 | + 1.06i·2-s − i·3-s − 0.128·4-s + 1.06·6-s + 0.925i·8-s − 9-s + 0.128i·12-s − 1.11·16-s − 0.252i·17-s − 1.06i·18-s + 1.56·19-s + 1.96i·23-s + 0.925·24-s + i·27-s + 1.98·31-s − 0.256i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.38932 + 1.38932i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38932 + 1.38932i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 81iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 17iT - 256T^{2} \) |
| 7 | \( 1 + 5.76e6T^{2} \) |
| 11 | \( 1 - 2.14e8T^{2} \) |
| 13 | \( 1 + 8.15e8T^{2} \) |
| 17 | \( 1 + 2.11e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 2.03e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 5.50e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.83e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 3.51e12T^{2} \) |
| 41 | \( 1 - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.16e13T^{2} \) |
| 47 | \( 1 - 8.06e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.26e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.43e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 4.06e14T^{2} \) |
| 71 | \( 1 - 6.45e14T^{2} \) |
| 73 | \( 1 + 8.06e14T^{2} \) |
| 79 | \( 1 - 6.96e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 3.84e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.93e15T^{2} \) |
| 97 | \( 1 + 7.83e15T^{2} \) |
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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
−13.54420653410565103323398183751, −12.03463020386680539693085394798, −11.28625091630429291855988155624, −9.425374975517169057412823621140, −8.020341811852842834617252662091, −7.35988039820324632836249891454, −6.24385332469594188633991352750, −5.22683797328678044307727658321, −2.88730279620532147633317227312, −1.26365004012369579093150663173,
0.66897873447030978731638254418, 2.49637904898507429020512654708, 3.60242279634992969322859654286, 4.88738845624000181590951079640, 6.54040448354097979665897131091, 8.374432549960790948234631982260, 9.712297620810837928519282221089, 10.36121985185948969219155767194, 11.42103550288830668724248085958, 12.20709965402453296629993409195