| L(s) = 1 | + 3.02·2-s − 118.·4-s − 1.50e3·7-s − 746.·8-s − 1.59e3·11-s − 956.·13-s − 4.54e3·14-s + 1.29e4·16-s − 3.24e4·17-s − 3.91e4·19-s − 4.82e3·22-s − 5.93e4·23-s − 2.88e3·26-s + 1.78e5·28-s − 6.61e4·29-s − 1.96e4·31-s + 1.34e5·32-s − 9.81e4·34-s + 3.76e5·37-s − 1.18e5·38-s − 3.85e5·41-s + 4.66e5·43-s + 1.89e5·44-s − 1.79e5·46-s − 4.68e5·47-s + 1.44e6·49-s + 1.13e5·52-s + ⋯ |
| L(s) = 1 | + 0.267·2-s − 0.928·4-s − 1.65·7-s − 0.515·8-s − 0.361·11-s − 0.120·13-s − 0.443·14-s + 0.790·16-s − 1.60·17-s − 1.31·19-s − 0.0966·22-s − 1.01·23-s − 0.0322·26-s + 1.54·28-s − 0.503·29-s − 0.118·31-s + 0.726·32-s − 0.428·34-s + 1.22·37-s − 0.349·38-s − 0.872·41-s + 0.894·43-s + 0.335·44-s − 0.271·46-s − 0.658·47-s + 1.75·49-s + 0.112·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3139560652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3139560652\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 3.02T + 128T^{2} \) |
| 7 | \( 1 + 1.50e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.59e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 956.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.24e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.91e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.93e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 6.61e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.96e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.76e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.85e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.66e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.68e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.60e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.04e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.78e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.64e3T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.79e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.01e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.76e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.06e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.14e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.02e6T + 8.07e13T^{2} \) |
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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
−10.85677888502692233002129065713, −9.833265823817668716997307782566, −9.173737148118689000300442846622, −8.182529261863423276794183800571, −6.68041571765823636005106248783, −5.94156401367573675085790794932, −4.53934618322383492108292957123, −3.64813931473471836341722823287, −2.38916984386249150520967279973, −0.26306418989210431745766180537,
0.26306418989210431745766180537, 2.38916984386249150520967279973, 3.64813931473471836341722823287, 4.53934618322383492108292957123, 5.94156401367573675085790794932, 6.68041571765823636005106248783, 8.182529261863423276794183800571, 9.173737148118689000300442846622, 9.833265823817668716997307782566, 10.85677888502692233002129065713
