| L(s) = 1 | + 27·3-s + 494.·5-s − 343·7-s + 729·9-s − 47.0·11-s + 3.62e3·13-s + 1.33e4·15-s − 1.46e4·17-s + 3.39e3·19-s − 9.26e3·21-s + 1.79e4·23-s + 1.66e5·25-s + 1.96e4·27-s + 1.39e5·29-s − 2.28e5·31-s − 1.27e3·33-s − 1.69e5·35-s + 4.38e5·37-s + 9.78e4·39-s + 3.12e5·41-s + 5.56e5·43-s + 3.60e5·45-s + 7.94e5·47-s + 1.17e5·49-s − 3.95e5·51-s + 2.04e6·53-s − 2.32e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.76·5-s − 0.377·7-s + 0.333·9-s − 0.0106·11-s + 0.457·13-s + 1.02·15-s − 0.722·17-s + 0.113·19-s − 0.218·21-s + 0.308·23-s + 2.12·25-s + 0.192·27-s + 1.06·29-s − 1.37·31-s − 0.00615·33-s − 0.668·35-s + 1.42·37-s + 0.264·39-s + 0.707·41-s + 1.06·43-s + 0.589·45-s + 1.11·47-s + 0.142·49-s − 0.417·51-s + 1.88·53-s − 0.0188·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.189120986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.189120986\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 + 343T \) |
| good | 5 | \( 1 - 494.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 47.0T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.62e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.46e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.39e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.79e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.39e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.28e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.38e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.12e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.56e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.94e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.04e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.56e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.46e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.15e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.38e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.97e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.17e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.09e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.16e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.40e6T + 8.07e13T^{2} \) |
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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
−10.23949043805861834624815463589, −9.246117207366631601223753420913, −8.912008533194302745710134198323, −7.44284867650974804212548408496, −6.36096259702579981444589320873, −5.67459658771807259343755777967, −4.37474437674183805216046217753, −2.91547427137788986405646992777, −2.10954403429676233105267740611, −0.994812049749729065156844147724,
0.994812049749729065156844147724, 2.10954403429676233105267740611, 2.91547427137788986405646992777, 4.37474437674183805216046217753, 5.67459658771807259343755777967, 6.36096259702579981444589320873, 7.44284867650974804212548408496, 8.912008533194302745710134198323, 9.246117207366631601223753420913, 10.23949043805861834624815463589
