| L(s) = 1 | + 27i·3-s + 69.8i·5-s − 860.·7-s − 729·9-s + 6.10e3i·11-s + 5.33e3i·13-s − 1.88e3·15-s + 3.86e4·17-s − 1.35e4i·19-s − 2.32e4i·21-s + 9.96e4·23-s + 7.32e4·25-s − 1.96e4i·27-s + 8.11e4i·29-s − 1.53e5·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.250i·5-s − 0.947·7-s − 0.333·9-s + 1.38i·11-s + 0.673i·13-s − 0.144·15-s + 1.90·17-s − 0.454i·19-s − 0.547i·21-s + 1.70·23-s + 0.937·25-s − 0.192i·27-s + 0.618i·29-s − 0.922·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.867219023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.867219023\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 69.8iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 860.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.10e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 5.33e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 3.86e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.35e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 9.96e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 8.11e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.53e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.67e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 3.84e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.50e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 5.03e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.61e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.58e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 2.61e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 1.52e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 1.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.52e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.64e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.34e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 1.70e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.70e6T + 8.07e13T^{2} \) |
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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
−10.40797211275729278637923874788, −9.499211803182691400735519695412, −9.082119112391717361607726212905, −7.47038675253156326437674339079, −6.89038254508606457601313644966, −5.63504478662689904950375874839, −4.65749996274850034986252841283, −3.52902455088443271614249950111, −2.62683894835042969262415584923, −1.08945284774274772324724056478,
0.48171075196552471783209751037, 1.15744366707027802686405694912, 2.95683447553892169693828823473, 3.44327635699956473402348654368, 5.27672079148910586569931795272, 5.94571112568097615654516326343, 6.97857638986938713385823279437, 7.984996616568574480383527229334, 8.784064939092455336426395087699, 9.802050861372486300568899947647
