| L(s) = 1 | + (3.09 + 5.35i)2-s + (44.8 − 77.6i)4-s + (−167. + 290. i)5-s + (442. + 765. i)7-s + 1.34e3·8-s − 2.07e3·10-s + (2.10e3 + 3.64e3i)11-s + (−6.25e3 + 1.08e4i)13-s + (−2.73e3 + 4.73e3i)14-s + (−1.57e3 − 2.72e3i)16-s + 742.·17-s + 9.11e3·19-s + (1.50e4 + 2.60e4i)20-s + (−1.30e4 + 2.25e4i)22-s + (2.26e4 − 3.92e4i)23-s + ⋯ |
| L(s) = 1 | + (0.273 + 0.473i)2-s + (0.350 − 0.606i)4-s + (−0.600 + 1.04i)5-s + (0.487 + 0.843i)7-s + 0.930·8-s − 0.657·10-s + (0.477 + 0.826i)11-s + (−0.789 + 1.36i)13-s + (−0.266 + 0.461i)14-s + (−0.0959 − 0.166i)16-s + 0.0366·17-s + 0.304·19-s + (0.421 + 0.729i)20-s + (−0.261 + 0.452i)22-s + (0.388 − 0.672i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0254 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0254 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.35746 + 1.39249i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35746 + 1.39249i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-3.09 - 5.35i)T + (-64 + 110. i)T^{2} \) |
| 5 | \( 1 + (167. - 290. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-442. - 765. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-2.10e3 - 3.64e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (6.25e3 - 1.08e4i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 742.T + 4.10e8T^{2} \) |
| 19 | \( 1 - 9.11e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-2.26e4 + 3.92e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.72e4 + 2.99e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-1.38e5 + 2.40e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 2.09e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-5.34e4 + 9.26e4i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (8.51e3 + 1.47e4i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-6.75e5 - 1.17e6i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.83e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-4.35e5 + 7.54e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (4.87e5 + 8.44e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.43e5 + 2.48e5i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 9.67e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.50e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (1.22e6 + 2.12e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-6.95e5 - 1.20e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 7.88e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-3.43e6 - 5.95e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−15.65822746078594989094095730745, −14.84161912489504879443661391184, −14.20289415358055502042031043059, −12.03127669033705223989134710335, −11.08978884996224484271255585690, −9.547370798646131913694693996477, −7.48636682524517636363520335638, −6.43232342209417304085745704183, −4.62304789085843018865209206816, −2.17344646477674092709092965650,
0.971517904559833672087795532531, 3.41820769535435579000164424120, 4.89262494199627318691227345441, 7.43052160023757267857233807536, 8.477300711322865576681896847936, 10.54623522694151407530356810996, 11.78223160942871805729763199024, 12.71665016087201505812700844739, 13.89607003301653411019984619518, 15.65064035198441768751150558655
