| L(s) = 1 | + 230. i·5-s − 3.84e3·7-s + 1.95e3i·11-s + 9.93e3·13-s − 1.12e5i·17-s + 7.54e4·19-s + 3.45e5i·23-s + 3.37e5·25-s + 1.34e6i·29-s − 1.51e6·31-s − 8.86e5i·35-s − 1.77e6·37-s + 3.33e6i·41-s − 7.13e4·43-s + 5.72e6i·47-s + ⋯ |
| L(s) = 1 | + 0.369i·5-s − 1.60·7-s + 0.133i·11-s + 0.347·13-s − 1.34i·17-s + 0.579·19-s + 1.23i·23-s + 0.863·25-s + 1.90i·29-s − 1.64·31-s − 0.590i·35-s − 0.945·37-s + 1.18i·41-s − 0.0208·43-s + 1.17i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.5992718085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5992718085\) |
| \(L(5)\) |
|
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 | \( 1 - 230. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 3.84e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.95e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 9.93e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.12e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 7.54e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 3.45e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 1.34e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.51e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.77e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.33e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 7.13e4T + 1.16e13T^{2} \) |
| 47 | \( 1 - 5.72e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 4.25e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 7.30e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.95e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 8.03e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 3.86e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.24e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 5.63e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + 7.31e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 4.03e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 2.17e7T + 7.83e15T^{2} \) |
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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
−9.417303470709256363244153462906, −9.077737054643744262422799581755, −7.43918999140554386315529317320, −6.93115524940523204950250990905, −5.93167701424944286431524293118, −4.92011610985846851119845979563, −3.30956133203406127081748845594, −3.10495710956699362870596398931, −1.45825432224840532622075349340, −0.15308412943508015779079051678,
0.73173403958300050984284672904, 2.14292286612781842633393503943, 3.34896703910948810059800715797, 4.09953300164527471970018626865, 5.53748237924960479222245338731, 6.30192564385125469403422273185, 7.15157218338153171418457856199, 8.426605768258544594107346460059, 9.100312148767565323999384847614, 10.07090700038723603092810896712
