| L(s) = 1 | − 76.0i·5-s + 222.·7-s + 1.90e3i·11-s − 309. i·13-s − 1.67e4·17-s − 2.70e4i·19-s + 7.73e4·23-s + 7.23e4·25-s + 1.56e5i·29-s − 2.65e5·31-s − 1.69e4i·35-s + 1.13e5i·37-s + 6.94e5·41-s − 9.00e5i·43-s − 7.71e4·47-s + ⋯ |
| L(s) = 1 | − 0.272i·5-s + 0.245·7-s + 0.431i·11-s − 0.0391i·13-s − 0.826·17-s − 0.904i·19-s + 1.32·23-s + 0.925·25-s + 1.19i·29-s − 1.60·31-s − 0.0668i·35-s + 0.367i·37-s + 1.57·41-s − 1.72i·43-s − 0.108·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.010486896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.010486896\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 76.0iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 222.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.90e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 309. iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.67e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.70e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 7.73e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.56e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 2.65e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.13e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 6.94e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.00e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 7.71e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.89e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 7.04e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.42e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 1.87e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 3.31e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.90e4T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.43e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.00e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 2.38e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.31e7T + 8.07e13T^{2} \) |
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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
−10.72177559243251116830275801925, −9.249264092603822017086613996978, −8.822899539884551519370608663197, −7.45091028998667295128725549166, −6.70985765629545035727088480046, −5.28692159626280867928474658334, −4.52831082051730581776653813737, −3.12738724280214682602436636903, −1.85613413155207105425835692460, −0.57340067135964850885436367454,
0.872014657887586458950348777585, 2.23044845350347229386726949713, 3.43724533560417104389101085023, 4.62530012528425310050535787987, 5.78399949944906059117978746618, 6.79702211357768576130448060311, 7.81977293390182414474994718234, 8.810012332458994583843397051600, 9.718267384273968315180369791831, 10.94327526426592931776868285531
