| L(s) = 1 | + 11.3i·2-s − 128.·4-s − 1.06e3i·5-s − 3.23e3·7-s − 1.44e3i·8-s + 1.20e4·10-s + 7.89e3i·11-s − 5.44e4·13-s − 3.65e4i·14-s + 1.63e4·16-s − 8.53e4i·17-s − 1.53e5·19-s + 1.35e5i·20-s − 8.92e4·22-s − 3.52e5i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 1.69i·5-s − 1.34·7-s − 0.353i·8-s + 1.20·10-s + 0.539i·11-s − 1.90·13-s − 0.952i·14-s + 0.250·16-s − 1.02i·17-s − 1.18·19-s + 0.849i·20-s − 0.381·22-s − 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.3902221555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3902221555\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 11.3iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.06e3iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 3.23e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 7.89e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 5.44e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 8.53e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.53e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 3.52e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 8.80e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.43e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.22e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.91e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.30e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 1.84e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 3.72e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 3.50e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 4.73e4T + 1.91e14T^{2} \) |
| 67 | \( 1 + 8.18e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 6.23e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.16e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 2.77e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 8.77e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 8.43e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 2.33e7T + 7.83e15T^{2} \) |
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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
−12.27066312841429554850722808194, −10.10264140051456963570277153302, −9.427862233153967284528957705660, −8.627660114405659624481461539847, −7.39305419916707843479801882565, −6.37088974570380854316878362355, −4.99935320880517234116772502099, −4.43329902519457220339823658467, −2.55137787745017680312204722275, −0.65684166529317716810486083706,
0.14843035047486259924333793619, 2.32445794704013231947651634398, 2.99068925370474074380008724316, 4.06858272618333753978385298087, 5.96229463722236185369310667001, 6.77874493307719742186089046624, 7.930634282995350371482223096308, 9.635217624546566408843969202044, 10.07608873383711732844836696963, 10.99024938166055465688325427553
