| L(s) = 1 | − 79.3·3-s + 336.·5-s + 343·7-s + 4.11e3·9-s + 7.30e3·11-s − 5.18e3·13-s − 2.66e4·15-s − 2.32e4·17-s + 1.08e4·19-s − 2.72e4·21-s + 3.37e4·23-s + 3.50e4·25-s − 1.52e5·27-s − 1.86e5·29-s − 9.81e4·31-s − 5.79e5·33-s + 1.15e5·35-s − 2.83e5·37-s + 4.11e5·39-s − 2.41e5·41-s − 7.47e5·43-s + 1.38e6·45-s + 1.01e6·47-s + 1.17e5·49-s + 1.84e6·51-s − 2.17e5·53-s + 2.45e6·55-s + ⋯ |
| L(s) = 1 | − 1.69·3-s + 1.20·5-s + 0.377·7-s + 1.88·9-s + 1.65·11-s − 0.654·13-s − 2.04·15-s − 1.14·17-s + 0.364·19-s − 0.641·21-s + 0.578·23-s + 0.448·25-s − 1.49·27-s − 1.41·29-s − 0.591·31-s − 2.80·33-s + 0.454·35-s − 0.919·37-s + 1.11·39-s − 0.547·41-s − 1.43·43-s + 2.26·45-s + 1.42·47-s + 0.142·49-s + 1.94·51-s − 0.201·53-s + 1.99·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 3 | \( 1 + 79.3T + 2.18e3T^{2} \) |
| 5 | \( 1 - 336.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 7.30e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.18e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.32e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.08e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.37e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.86e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.81e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.83e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.41e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.47e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.01e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.17e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.04e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.16e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.32e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.28e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.59e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.40e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.37e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.19e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.35e6T + 8.07e13T^{2} \) |
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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
−9.624003519853083341007378656574, −8.915436949792175498079988675892, −7.09944521098052179957041425587, −6.60550566363698863238703809762, −5.66632557466656215938270631927, −5.04225982950834502810831613901, −3.95631905623942876787734838609, −1.99819974231335798518042126430, −1.24338517804061481486133146851, 0,
1.24338517804061481486133146851, 1.99819974231335798518042126430, 3.95631905623942876787734838609, 5.04225982950834502810831613901, 5.66632557466656215938270631927, 6.60550566363698863238703809762, 7.09944521098052179957041425587, 8.915436949792175498079988675892, 9.624003519853083341007378656574
