L(s) = 1 | − 324. i·5-s + 956.·7-s + 5.45e3i·11-s − 6.28e3i·13-s − 3.45e4·17-s + 1.45e4i·19-s − 2.46e4·23-s − 2.71e4·25-s − 1.71e5i·29-s − 1.11e5·31-s − 3.10e5i·35-s − 1.03e5i·37-s − 7.16e4·41-s + 3.28e5i·43-s + 1.19e5·47-s + ⋯ |
L(s) = 1 | − 1.16i·5-s + 1.05·7-s + 1.23i·11-s − 0.793i·13-s − 1.70·17-s + 0.488i·19-s − 0.422·23-s − 0.347·25-s − 1.30i·29-s − 0.673·31-s − 1.22i·35-s − 0.336i·37-s − 0.162·41-s + 0.629i·43-s + 0.167·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.03319267175\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03319267175\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 324. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 956.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.45e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 6.28e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 3.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.45e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 2.46e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.71e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.11e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.03e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 7.16e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.28e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 1.19e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.04e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 2.25e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 1.55e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 3.16e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 5.38e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.22e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.89e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 4.37e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.84e6T + 8.07e13T^{2} \) |
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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
−9.971958285575521268507654097567, −8.966267380438181092691916257170, −8.184694959668858853792734069793, −7.31393585123260062583041186088, −5.86251957187990376368455940599, −4.76576137186453328773351532869, −4.25350040481324500070445983841, −2.27413555065958061017605826389, −1.35559526038004661783258325906, −0.00693295924804094371925221661,
1.66084392326879090507353123228, 2.74905175919918062335940748359, 3.96058232578594535571834582775, 5.14771507170683438146214607249, 6.44220810204391801722733248322, 7.11099124764168262966228790107, 8.362108303672253250773397974997, 9.077158143003600146507266343876, 10.54234957680665203228118903201, 11.15350406965528913200002403367