| L(s) = 1 | + 20.4·3-s − 21.4·5-s + 49·7-s + 173.·9-s − 647.·11-s + 787.·13-s − 438.·15-s − 1.93e3·17-s + 964.·19-s + 1.00e3·21-s + 1.49e3·23-s − 2.66e3·25-s − 1.41e3·27-s − 7.16e3·29-s + 2.22e3·31-s − 1.32e4·33-s − 1.05e3·35-s + 2.05e3·37-s + 1.60e4·39-s + 9.00e3·41-s + 2.60e3·43-s − 3.73e3·45-s − 2.22e4·47-s + 2.40e3·49-s − 3.95e4·51-s − 2.77e4·53-s + 1.39e4·55-s + ⋯ |
| L(s) = 1 | + 1.30·3-s − 0.384·5-s + 0.377·7-s + 0.715·9-s − 1.61·11-s + 1.29·13-s − 0.503·15-s − 1.62·17-s + 0.613·19-s + 0.495·21-s + 0.588·23-s − 0.852·25-s − 0.372·27-s − 1.58·29-s + 0.415·31-s − 2.11·33-s − 0.145·35-s + 0.246·37-s + 1.69·39-s + 0.837·41-s + 0.214·43-s − 0.275·45-s − 1.46·47-s + 0.142·49-s − 2.12·51-s − 1.35·53-s + 0.620·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 20.4T + 243T^{2} \) |
| 5 | \( 1 + 21.4T + 3.12e3T^{2} \) |
| 11 | \( 1 + 647.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 787.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.93e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 964.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.49e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.16e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.22e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.05e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.00e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.60e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.22e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.77e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.38e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.14e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.41e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.87e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.55e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.13e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.35e5T + 8.58e9T^{2} \) |
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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
−9.583838549343828307305654377737, −8.823585522904323171392899580840, −8.003083378764929146865164086637, −7.53747911734611961576564283520, −6.09820340466253801895323243102, −4.83734545923323659645603526439, −3.71939547140765974438537540443, −2.78015990291136604824428568899, −1.75847346102821294150853831875, 0,
1.75847346102821294150853831875, 2.78015990291136604824428568899, 3.71939547140765974438537540443, 4.83734545923323659645603526439, 6.09820340466253801895323243102, 7.53747911734611961576564283520, 8.003083378764929146865164086637, 8.823585522904323171392899580840, 9.583838549343828307305654377737
