| L(s) = 1 | + 11.3i·2-s − 128.·4-s + 618. i·5-s − 549.·7-s − 1.44e3i·8-s − 6.99e3·10-s − 1.92e4i·11-s + 2.08e3·13-s − 6.21e3i·14-s + 1.63e4·16-s + 5.81e4i·17-s + 2.02e5·19-s − 7.91e4i·20-s + 2.18e5·22-s − 6.90e4i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 0.988i·5-s − 0.228·7-s − 0.353i·8-s − 0.699·10-s − 1.31i·11-s + 0.0729·13-s − 0.161i·14-s + 0.250·16-s + 0.696i·17-s + 1.55·19-s − 0.494i·20-s + 0.930·22-s − 0.246i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.897046487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.897046487\) |
| \(L(5)\) |
|
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 | \( 1 - 618. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 549.T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.92e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.08e3T + 8.15e8T^{2} \) |
| 17 | \( 1 - 5.81e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 2.02e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 6.90e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 3.62e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 5.45e4T + 8.52e11T^{2} \) |
| 37 | \( 1 - 6.32e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 5.05e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.99e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 7.04e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 4.75e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 6.52e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 9.32e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.17e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 3.81e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 5.36e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 5.15e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 3.30e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 1.36e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.10e8T + 7.83e15T^{2} \) |
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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
−11.44089692340248231901537746312, −10.60286392365932041563978921554, −9.496221909515839305877392581099, −8.365079008873366626076999873556, −7.35231383465753443555887106262, −6.33699161860513947077346448981, −5.50165470093472221681167620115, −3.82029644935601849876109611816, −2.82393714169690310587027056598, −0.844315332310920484888870373235,
0.66397986049757119928875975777, 1.73063820956850096207760301638, 3.16305467996213042064260113876, 4.56826234334183285500742062603, 5.30369873355272299932793627749, 7.01799187556822074483554460075, 8.181462419215582459658257981670, 9.442429598048949868859524661919, 9.810920789148714991045170878199, 11.28671104385877457718756459998
