| L(s) = 1 | − 17.0·2-s − 81·3-s − 220.·4-s + 625·5-s + 1.38e3·6-s + 1.18e4·7-s + 1.25e4·8-s + 6.56e3·9-s − 1.06e4·10-s + 1.07e4·11-s + 1.78e4·12-s + 5.58e4·13-s − 2.01e5·14-s − 5.06e4·15-s − 1.00e5·16-s + 3.90e4·17-s − 1.12e5·18-s + 1.30e5·19-s − 1.37e5·20-s − 9.56e5·21-s − 1.83e5·22-s + 8.55e4·23-s − 1.01e6·24-s + 3.90e5·25-s − 9.52e5·26-s − 5.31e5·27-s − 2.60e6·28-s + ⋯ |
| L(s) = 1 | − 0.754·2-s − 0.577·3-s − 0.430·4-s + 0.447·5-s + 0.435·6-s + 1.85·7-s + 1.07·8-s + 0.333·9-s − 0.337·10-s + 0.220·11-s + 0.248·12-s + 0.541·13-s − 1.40·14-s − 0.258·15-s − 0.383·16-s + 0.113·17-s − 0.251·18-s + 0.229·19-s − 0.192·20-s − 1.07·21-s − 0.166·22-s + 0.0637·23-s − 0.623·24-s + 0.200·25-s − 0.408·26-s − 0.192·27-s − 0.800·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 19 | \( 1 - 1.30e5T \) |
| good | 2 | \( 1 + 17.0T + 512T^{2} \) |
| 7 | \( 1 - 1.18e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.07e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.58e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.90e4T + 1.18e11T^{2} \) |
| 23 | \( 1 - 8.55e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.91e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.66e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.74e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.56e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.52e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 9.69e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.79e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 2.56e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.01e6T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.24e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 6.68e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.31e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.81e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.01e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.69e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.63e8T + 7.60e17T^{2} \) |
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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
−9.704813099796008588389995210895, −8.825992070640049593279317546746, −8.004099496012807337203738480406, −7.13901022585534496000111186533, −5.59348735833076272188069085023, −4.95455816119983495697684558294, −3.89328824620782590938347365067, −1.76452127198963063999542900463, −1.32761561269559891038024046584, 0,
1.32761561269559891038024046584, 1.76452127198963063999542900463, 3.89328824620782590938347365067, 4.95455816119983495697684558294, 5.59348735833076272188069085023, 7.13901022585534496000111186533, 8.004099496012807337203738480406, 8.825992070640049593279317546746, 9.704813099796008588389995210895
