| L(s) = 1 | − 9.17·2-s + 65.7·3-s − 427.·4-s + 625·5-s − 602.·6-s + 2.40e3·7-s + 8.62e3·8-s − 1.53e4·9-s − 5.73e3·10-s + 3.50e4·11-s − 2.81e4·12-s − 7.74e4·13-s − 2.20e4·14-s + 4.10e4·15-s + 1.40e5·16-s − 2.29e5·17-s + 1.40e5·18-s + 1.64e4·19-s − 2.67e5·20-s + 1.57e5·21-s − 3.21e5·22-s − 2.57e6·23-s + 5.66e5·24-s + 3.90e5·25-s + 7.09e5·26-s − 2.30e6·27-s − 1.02e6·28-s + ⋯ |
| L(s) = 1 | − 0.405·2-s + 0.468·3-s − 0.835·4-s + 0.447·5-s − 0.189·6-s + 0.377·7-s + 0.744·8-s − 0.780·9-s − 0.181·10-s + 0.722·11-s − 0.391·12-s − 0.751·13-s − 0.153·14-s + 0.209·15-s + 0.534·16-s − 0.667·17-s + 0.316·18-s + 0.0289·19-s − 0.373·20-s + 0.177·21-s − 0.292·22-s − 1.91·23-s + 0.348·24-s + 0.200·25-s + 0.304·26-s − 0.834·27-s − 0.315·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 + 9.17T + 512T^{2} \) |
| 3 | \( 1 - 65.7T + 1.96e4T^{2} \) |
| 11 | \( 1 - 3.50e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.74e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.29e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.64e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.57e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.62e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.17e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.70e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.98e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.95e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.93e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.74e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 5.24e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.23e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.74e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.63e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.09e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.65e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 9.43e6T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.64e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.20e9T + 7.60e17T^{2} \) |
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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
−14.13385977967800453904270733142, −12.90639181970161657491855120227, −11.31771657356272869470115669982, −9.731665429943266820692532590976, −8.905595636040902495495870369867, −7.68905770628825976643235786050, −5.66046106946367216334775453763, −4.02160144344251659376974537382, −1.97470375620916312201725157159, 0,
1.97470375620916312201725157159, 4.02160144344251659376974537382, 5.66046106946367216334775453763, 7.68905770628825976643235786050, 8.905595636040902495495870369867, 9.731665429943266820692532590976, 11.31771657356272869470115669982, 12.90639181970161657491855120227, 14.13385977967800453904270733142
