| L(s) = 1 | + 1.11e3i·5-s + 285.·7-s + 2.21e4i·11-s − 3.84e4·13-s − 4.41e4i·17-s − 6.00e4·19-s + 2.89e5i·23-s − 8.55e5·25-s − 9.03e5i·29-s − 1.20e5·31-s + 3.18e5i·35-s − 2.67e6·37-s − 2.54e6i·41-s + 3.26e6·43-s + 9.69e5i·47-s + ⋯ |
| L(s) = 1 | + 1.78i·5-s + 0.118·7-s + 1.51i·11-s − 1.34·13-s − 0.528i·17-s − 0.461·19-s + 1.03i·23-s − 2.19·25-s − 1.27i·29-s − 0.129·31-s + 0.212i·35-s − 1.42·37-s − 0.902i·41-s + 0.954·43-s + 0.198i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.2125550952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2125550952\) |
| \(L(5)\) |
|
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 | \( 1 - 1.11e3iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 285.T + 5.76e6T^{2} \) |
| 11 | \( 1 - 2.21e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.84e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 4.41e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 6.00e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 2.89e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 9.03e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.20e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.67e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 2.54e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 3.26e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 9.69e5iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.46e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 2.83e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.03e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 3.39e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 7.67e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.26e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 6.59e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 1.75e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 8.38e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.15e8T + 7.83e15T^{2} \) |
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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
−10.80406303242041297060349448305, −10.02614115073108065386260575472, −9.415221796809190324136199734053, −7.57884219111941284748611381798, −7.31123179622319059132133934433, −6.35136842824487178579287300397, −5.04434681147497821085779710720, −3.87429726202467499767408931418, −2.63175520755759583593173342304, −2.00275166007086312073887042956,
0.04940228754760040067426460507, 0.852229231062970287571024773968, 2.01341124605180104520993692086, 3.52818771670759771946365767942, 4.76131032959412641436553422263, 5.33100224682776416662148503002, 6.51352121123322101689262590407, 7.987256059256749100944379703034, 8.577549479876785577238154223877, 9.300949663142568383877316885161
