| L(s) = 1 | + 27i·3-s + 124. i·5-s + 1.16e3·7-s − 729·9-s + 2.26e3i·11-s − 3.02e3i·13-s − 3.35e3·15-s − 1.81e4·17-s − 3.32e4i·19-s + 3.14e4i·21-s + 8.58e4·23-s + 6.27e4·25-s − 1.96e4i·27-s − 1.58e5i·29-s + 1.99e5·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.443i·5-s + 1.28·7-s − 0.333·9-s + 0.513i·11-s − 0.382i·13-s − 0.256·15-s − 0.894·17-s − 1.11i·19-s + 0.741i·21-s + 1.47·23-s + 0.802·25-s − 0.192i·27-s − 1.20i·29-s + 1.20·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.565103856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.565103856\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27iT \) |
| good | 5 | \( 1 - 124. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 1.16e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.26e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 3.02e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.81e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.32e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 8.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.58e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 1.99e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.54e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 2.18e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.48e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 7.03e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.18e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.97e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 3.14e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 1.03e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.24e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.36e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.65e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.63e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 2.71e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.27e7T + 8.07e13T^{2} \) |
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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
−10.29626774658056647588447792328, −9.203798965937579634869029863034, −8.399222109805919639672337817653, −7.38598779104150745979943480936, −6.41474642686554418768855848409, −4.94371366615398915590469874453, −4.59751439187557971702107691197, −3.09078117289678029929511384004, −2.06182572071716384863532798639, −0.61994062578148757601977939441,
0.981829403953599644520794780003, 1.66103551849371559941311645875, 2.99940728793689597838943475506, 4.49885057786072140354726320878, 5.23545319029967391261739069536, 6.44882411828112185417193634760, 7.39908093482210257035770796116, 8.503296955757780527063555926047, 8.782226807184229341985298990700, 10.29832183223309255866950507443
