| L(s) = 1 | + 8i·2-s + 27i·3-s − 64·4-s − 216·6-s − 988i·7-s − 512i·8-s − 729·9-s − 8.04e3·11-s − 1.72e3i·12-s + 3.33e3i·13-s + 7.90e3·14-s + 4.09e3·16-s + 6.58e3i·17-s − 5.83e3i·18-s + 2.74e4·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.08i·7-s − 0.353i·8-s − 0.333·9-s − 1.82·11-s − 0.288i·12-s + 0.420i·13-s + 0.769·14-s + 0.250·16-s + 0.324i·17-s − 0.235i·18-s + 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.521820546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.521820546\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8iT \) |
| 3 | \( 1 - 27iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 988iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 8.04e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.33e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 6.58e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 2.74e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.86e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 1.32e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.54e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.19e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 9.23e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.34e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 1.27e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 8.35e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 3.06e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.00e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.08e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 3.66e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.12e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 4.12e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.58e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 5.76e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.74e6iT - 8.07e13T^{2} \) |
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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
−11.82957637346468736596316953398, −10.37157253146118047905143806682, −10.11963302022571021446296684233, −8.535409816091331914409079973231, −7.70981577926883376693814812200, −6.57713362165829208589477584732, −5.19367120382675556141997339137, −4.34175353228633845951893783581, −2.89242788396088399298123728289, −0.69448766416874126982960865037,
0.68633964217214723757157626426, 2.29608609106302129126504774458, 3.02409600981238181618559015111, 5.02039526231186081966914078584, 5.81936462114732489458473556228, 7.54266166111324126413640217316, 8.378154957688452958374154989317, 9.551958405922810777014205554332, 10.59811131811901429942772406552, 11.66950331044554007293304700140
