| L(s) = 1 | − 3.72·2-s + 21.0·3-s − 18.1·4-s − 25·5-s − 78.4·6-s + 186.·8-s + 200.·9-s + 93.1·10-s − 164.·11-s − 381.·12-s − 250.·13-s − 526.·15-s − 116.·16-s + 2.17e3·17-s − 745.·18-s + 18.8·19-s + 452.·20-s + 613.·22-s − 2.62e3·23-s + 3.93e3·24-s + 625·25-s + 932.·26-s − 905.·27-s − 7.25e3·29-s + 1.96e3·30-s + 1.68e3·31-s − 5.54e3·32-s + ⋯ |
| L(s) = 1 | − 0.658·2-s + 1.35·3-s − 0.566·4-s − 0.447·5-s − 0.889·6-s + 1.03·8-s + 0.823·9-s + 0.294·10-s − 0.410·11-s − 0.764·12-s − 0.410·13-s − 0.603·15-s − 0.113·16-s + 1.82·17-s − 0.542·18-s + 0.0119·19-s + 0.253·20-s + 0.270·22-s − 1.03·23-s + 1.39·24-s + 0.200·25-s + 0.270·26-s − 0.238·27-s − 1.60·29-s + 0.397·30-s + 0.315·31-s − 0.956·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.72T + 32T^{2} \) |
| 3 | \( 1 - 21.0T + 243T^{2} \) |
| 11 | \( 1 + 164.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 250.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.17e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 18.8T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.62e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.25e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 738.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.56e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.86e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.48e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.26e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.48e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.09e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.30e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.13e5T + 8.58e9T^{2} \) |
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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
−10.22632305870171322911685620338, −9.689039854950306854840129227426, −8.711624410054702221732056836356, −7.912560777980995649257187260030, −7.46931570779025687343815375569, −5.47110177389975171486060674025, −4.09239292095277801017603742711, −3.11554212127641276565756785296, −1.61526666963171641436844138913, 0,
1.61526666963171641436844138913, 3.11554212127641276565756785296, 4.09239292095277801017603742711, 5.47110177389975171486060674025, 7.46931570779025687343815375569, 7.912560777980995649257187260030, 8.711624410054702221732056836356, 9.689039854950306854840129227426, 10.22632305870171322911685620338
