| L(s) = 1 | − 25.7·3-s + 34.4·5-s − 49·7-s + 421.·9-s − 783.·11-s + 894.·13-s − 888.·15-s + 237.·17-s − 1.77e3·19-s + 1.26e3·21-s + 4.26e3·23-s − 1.93e3·25-s − 4.61e3·27-s − 5.26e3·29-s + 7.51e3·31-s + 2.02e4·33-s − 1.68e3·35-s − 676.·37-s − 2.30e4·39-s + 5.94e3·41-s + 2.23e4·43-s + 1.45e4·45-s + 1.16e4·47-s + 2.40e3·49-s − 6.12e3·51-s + 1.01e4·53-s − 2.70e4·55-s + ⋯ |
| L(s) = 1 | − 1.65·3-s + 0.616·5-s − 0.377·7-s + 1.73·9-s − 1.95·11-s + 1.46·13-s − 1.02·15-s + 0.199·17-s − 1.12·19-s + 0.625·21-s + 1.68·23-s − 0.619·25-s − 1.21·27-s − 1.16·29-s + 1.40·31-s + 3.23·33-s − 0.233·35-s − 0.0812·37-s − 2.42·39-s + 0.552·41-s + 1.84·43-s + 1.07·45-s + 0.770·47-s + 0.142·49-s − 0.329·51-s + 0.494·53-s − 1.20·55-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 | \( 1 + 25.7T + 243T^{2} \) |
| 5 | \( 1 - 34.4T + 3.12e3T^{2} \) |
| 11 | \( 1 + 783.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 894.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 237.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.77e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.26e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 676.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.23e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.16e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.01e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.20e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.52e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.05e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.46e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.65e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.32e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.64e4T + 8.58e9T^{2} \) |
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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
−10.28860228461224415686728843449, −9.099178920478421579487979945253, −7.85993137912303200417614049603, −6.73412758051757377625300201160, −5.84246942874584288408842423714, −5.43009669132023032022093143943, −4.24713750846946193031953427117, −2.62061380376149514889426432387, −1.09306588344719249778953607354, 0,
1.09306588344719249778953607354, 2.62061380376149514889426432387, 4.24713750846946193031953427117, 5.43009669132023032022093143943, 5.84246942874584288408842423714, 6.73412758051757377625300201160, 7.85993137912303200417614049603, 9.099178920478421579487979945253, 10.28860228461224415686728843449
