| L(s) = 1 | − 43.7·2-s + 81·3-s + 1.40e3·4-s + 625·5-s − 3.54e3·6-s + 6.87e3·7-s − 3.90e4·8-s + 6.56e3·9-s − 2.73e4·10-s − 6.93e4·11-s + 1.13e5·12-s − 6.71e4·13-s − 3.00e5·14-s + 5.06e4·15-s + 9.91e5·16-s + 8.27e4·17-s − 2.87e5·18-s − 1.30e5·19-s + 8.77e5·20-s + 5.56e5·21-s + 3.03e6·22-s − 6.73e5·23-s − 3.16e6·24-s + 3.90e5·25-s + 2.93e6·26-s + 5.31e5·27-s + 9.65e6·28-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 0.577·3-s + 2.74·4-s + 0.447·5-s − 1.11·6-s + 1.08·7-s − 3.37·8-s + 0.333·9-s − 0.865·10-s − 1.42·11-s + 1.58·12-s − 0.651·13-s − 2.09·14-s + 0.258·15-s + 3.78·16-s + 0.240·17-s − 0.644·18-s − 0.229·19-s + 1.22·20-s + 0.624·21-s + 2.76·22-s − 0.501·23-s − 1.94·24-s + 0.200·25-s + 1.26·26-s + 0.192·27-s + 2.96·28-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 | \( 1 + 43.7T + 512T^{2} \) |
| 7 | \( 1 - 6.87e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 6.93e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.71e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 8.27e4T + 1.18e11T^{2} \) |
| 23 | \( 1 + 6.73e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.65e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.82e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.79e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.13e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.52e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.06e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.12e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.42e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.46e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.06e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 9.20e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.20e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.50e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.11e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.80e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.42e9T + 7.60e17T^{2} \) |
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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
−9.878169726178926685330010590297, −8.681514816219703518697369694162, −8.026233222292423583493381047016, −7.50186113734890236135629408528, −6.26236292276161686000658825995, −4.94276040950617840904630921026, −2.84591425966715379185201049786, −2.16701079471778177707478521195, −1.22299144287927450051948131199, 0,
1.22299144287927450051948131199, 2.16701079471778177707478521195, 2.84591425966715379185201049786, 4.94276040950617840904630921026, 6.26236292276161686000658825995, 7.50186113734890236135629408528, 8.026233222292423583493381047016, 8.681514816219703518697369694162, 9.878169726178926685330010590297
