| L(s) = 1 | + 11.3i·2-s − 128.·4-s − 197. i·5-s + 907.·7-s − 1.44e3i·8-s + 2.23e3·10-s + 2.89e3i·11-s − 3.85e4·13-s + 1.02e4i·14-s + 1.63e4·16-s − 9.79e4i·17-s + 1.87e5·19-s + 2.53e4i·20-s − 3.27e4·22-s + 2.61e5i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 0.316i·5-s + 0.377·7-s − 0.353i·8-s + 0.223·10-s + 0.197i·11-s − 1.34·13-s + 0.267i·14-s + 0.250·16-s − 1.17i·17-s + 1.44·19-s + 0.158i·20-s − 0.139·22-s + 0.934i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.156724420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.156724420\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 11.3iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 907.T \) |
| good | 5 | \( 1 + 197. iT - 3.90e5T^{2} \) |
| 11 | \( 1 - 2.89e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.85e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 9.79e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.87e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 2.61e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 9.51e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.46e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 2.88e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.52e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 5.54e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 9.49e5iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 6.10e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 5.27e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 7.53e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 4.97e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 8.60e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.63e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 2.54e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 6.80e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.95e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 8.41e7T + 7.83e15T^{2} \) |
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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
−12.30000280808332212072514558555, −11.32123977407853652714312842054, −9.791630370801363619497504530583, −9.117649693261324848603868130071, −7.68137798563024839334256089031, −7.08171966705361832591607755614, −5.39050671729682171369067378187, −4.75624229019900749812488834148, −3.01746649031222672564726970764, −1.20887277875194436262700740237,
0.32685908073643346237088386429, 1.85100017826085901101308074902, 3.05011252005428137986192608850, 4.41558895655173956105271129661, 5.62018426396307026165007701320, 7.17527047490816791693507627008, 8.284948976820484173151491133120, 9.535194273185892366549604019827, 10.41837088972922666706670705954, 11.40970116092556003419175284555
