| L(s) = 1 | + 126. i·5-s + 2.83e3·7-s + 2.74e4i·11-s + 4.70e4·13-s + 1.47e5i·17-s + 1.94e5·19-s + 1.32e5i·23-s + 3.74e5·25-s + 3.67e5i·29-s + 1.81e5·31-s + 3.58e5i·35-s − 9.84e5·37-s + 3.24e6i·41-s − 9.39e5·43-s − 6.65e6i·47-s + ⋯ |
| L(s) = 1 | + 0.202i·5-s + 1.17·7-s + 1.87i·11-s + 1.64·13-s + 1.77i·17-s + 1.49·19-s + 0.472i·23-s + 0.959·25-s + 0.520i·29-s + 0.196·31-s + 0.238i·35-s − 0.525·37-s + 1.15i·41-s − 0.274·43-s − 1.36i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.359986137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.359986137\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 126. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 2.83e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 2.74e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 4.70e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.47e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.94e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.32e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 3.67e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.81e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 9.84e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.24e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 9.39e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + 6.65e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.20e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.84e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.24e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.55e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 4.32e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.01e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.66e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 6.93e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 7.27e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 7.28e7T + 7.83e15T^{2} \) |
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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
−10.19626776121486187461829372679, −9.062008766998412556604551745172, −8.170405517971273891113284934417, −7.39831330343754992555694393336, −6.37023472047898784938132192376, −5.23944303460757074493513482893, −4.36296589082084545461980596178, −3.32490572063772446926851344722, −1.72000976880548499940810621538, −1.35004485580398923026909225493,
0.71368869035331428366648177633, 1.17363023876483424524518026032, 2.78158980362298150435817787623, 3.69297043075237231847522886025, 5.01611530871264227862407151406, 5.65692272282716252350624676353, 6.81201914787269813714793296820, 8.020950965344667248050254066434, 8.564511100496807686618908353604, 9.396284966817378861486757804958
