| L(s) = 1 | + 28.1·3-s − 10.1·5-s − 135.·7-s + 550.·9-s + 74.1·11-s − 252.·13-s − 285.·15-s + 233.·17-s − 550.·19-s − 3.82e3·21-s − 1.95e3·23-s − 3.02e3·25-s + 8.67e3·27-s − 4.67e3·29-s − 3.33e3·31-s + 2.08e3·33-s + 1.37e3·35-s − 1.34e4·37-s − 7.11e3·39-s + 8.20e3·41-s + 1.84e3·43-s − 5.59e3·45-s + 2.20e4·47-s + 1.64e3·49-s + 6.58e3·51-s + 6.34e3·53-s − 752.·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s − 0.181·5-s − 1.04·7-s + 2.26·9-s + 0.184·11-s − 0.414·13-s − 0.328·15-s + 0.195·17-s − 0.349·19-s − 1.89·21-s − 0.769·23-s − 0.967·25-s + 2.29·27-s − 1.03·29-s − 0.622·31-s + 0.333·33-s + 0.190·35-s − 1.62·37-s − 0.748·39-s + 0.762·41-s + 0.152·43-s − 0.411·45-s + 1.45·47-s + 0.0980·49-s + 0.354·51-s + 0.310·53-s − 0.0335·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 3 | \( 1 - 28.1T + 243T^{2} \) |
| 5 | \( 1 + 10.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 135.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 74.1T + 1.61e5T^{2} \) |
| 13 | \( 1 + 252.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 233.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 550.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.95e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.67e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.33e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.34e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.20e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 2.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.34e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.49e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.91e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.26e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.23e4T + 8.58e9T^{2} \) |
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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
−9.291327187623994596011625264125, −8.496159612875050748687522400835, −7.59504564702606254636325101854, −6.98845726970075256053335075781, −5.75582807887206089568283375852, −4.15331767259955527944696463651, −3.58064796194844698581458143700, −2.63074838619592954925060875540, −1.71271772375828603266921365869, 0,
1.71271772375828603266921365869, 2.63074838619592954925060875540, 3.58064796194844698581458143700, 4.15331767259955527944696463651, 5.75582807887206089568283375852, 6.98845726970075256053335075781, 7.59504564702606254636325101854, 8.496159612875050748687522400835, 9.291327187623994596011625264125
