| L(s) = 1 | + 1.15e3·5-s − 1.31e3i·7-s + 1.50e4i·11-s − 3.08e4·13-s − 4.64e4·17-s − 6.91e4i·19-s − 3.93e5i·23-s + 9.53e5·25-s + 7.74e5·29-s + 4.66e5i·31-s − 1.52e6i·35-s + 4.99e5·37-s + 3.81e6·41-s − 2.22e6i·43-s − 7.92e6i·47-s + ⋯ |
| L(s) = 1 | + 1.85·5-s − 0.546i·7-s + 1.02i·11-s − 1.08·13-s − 0.556·17-s − 0.530i·19-s − 1.40i·23-s + 2.44·25-s + 1.09·29-s + 0.505i·31-s − 1.01i·35-s + 0.266·37-s + 1.35·41-s − 0.651i·43-s − 1.62i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.075512166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.075512166\) |
| \(L(5)\) |
|
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 | \( 1 - 1.15e3T + 3.90e5T^{2} \) |
| 7 | \( 1 + 1.31e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.50e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.08e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 4.64e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 6.91e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 3.93e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 7.74e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 4.66e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 4.99e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.81e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 2.22e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 7.92e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.22e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 2.04e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.27e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.90e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 4.88e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.46e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 6.22e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 7.55e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 7.40e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 1.23e8T + 7.83e15T^{2} \) |
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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
−10.06981008642747332634512380798, −9.633134520897624932656257694478, −8.572937121856356314356707244046, −7.05686154049061123827010598573, −6.52188801258959715035950414645, −5.22514047684242375098399739048, −4.49740346260088827187466381627, −2.60218824374206489725040895662, −1.99945499467785884546383880112, −0.65976585300572750701032378591,
1.05307200783375313545300973679, 2.16968323316049784082128647037, 2.96866705447406588790764043211, 4.75936043662016396225464674971, 5.79470076925501564568660562155, 6.23375963646233718924843044044, 7.64331573273582188294609369967, 8.952022514817052055389704102366, 9.514927046077147092035231243930, 10.35166118317956793275227113520
