L(s) = 1 | + (−0.921 − 1.07i)2-s − 3-s + (−0.300 + 1.97i)4-s − 1.54i·5-s + (0.921 + 1.07i)6-s + 4.78·7-s + (2.39 − 1.50i)8-s + 9-s + (−1.65 + 1.42i)10-s + 3.05·11-s + (0.300 − 1.97i)12-s + 3.50i·13-s + (−4.40 − 5.12i)14-s + 1.54i·15-s + (−3.81 − 1.18i)16-s − 4.48·17-s + ⋯ |
L(s) = 1 | + (−0.651 − 0.758i)2-s − 0.577·3-s + (−0.150 + 0.988i)4-s − 0.691i·5-s + (0.376 + 0.437i)6-s + 1.80·7-s + (0.847 − 0.530i)8-s + 0.333·9-s + (−0.524 + 0.451i)10-s + 0.921·11-s + (0.0867 − 0.570i)12-s + 0.971i·13-s + (−1.17 − 1.37i)14-s + 0.399i·15-s + (−0.954 − 0.296i)16-s − 1.08·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10799 - 0.259808i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10799 - 0.259808i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.921 + 1.07i)T \) |
| 3 | \( 1 + T \) |
| 67 | \( 1 + (2.49 - 7.79i)T \) |
good | 5 | \( 1 + 1.54iT - 5T^{2} \) |
| 7 | \( 1 - 4.78T + 7T^{2} \) |
| 11 | \( 1 - 3.05T + 11T^{2} \) |
| 13 | \( 1 - 3.50iT - 13T^{2} \) |
| 17 | \( 1 + 4.48T + 17T^{2} \) |
| 19 | \( 1 - 6.47iT - 19T^{2} \) |
| 23 | \( 1 - 5.90iT - 23T^{2} \) |
| 29 | \( 1 - 5.56T + 29T^{2} \) |
| 31 | \( 1 + 8.11T + 31T^{2} \) |
| 37 | \( 1 - 2.86T + 37T^{2} \) |
| 41 | \( 1 + 5.82iT - 41T^{2} \) |
| 43 | \( 1 - 8.69T + 43T^{2} \) |
| 47 | \( 1 + 3.02iT - 47T^{2} \) |
| 53 | \( 1 + 5.29iT - 53T^{2} \) |
| 59 | \( 1 - 14.0iT - 59T^{2} \) |
| 61 | \( 1 + 0.366iT - 61T^{2} \) |
| 71 | \( 1 + 4.06iT - 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 7.10iT - 83T^{2} \) |
| 89 | \( 1 - 4.89T + 89T^{2} \) |
| 97 | \( 1 + 5.71iT - 97T^{2} \) |
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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
−10.40312930701617745142006453351, −9.141877158808022802439272864738, −8.800375507689634439477835772823, −7.80958448548773575734072767737, −7.00328198971430988055426772366, −5.60912724789033010935073872921, −4.51239678732247572593855925567, −4.00761837950874640602689548268, −1.91190040165403171099378037626, −1.28721531746563933478051822696,
0.942458577335231169673621104163, 2.36415472493996013899398591312, 4.47361131911997999855175954646, 4.98395230673110149703507953144, 6.17317580032660480961324654279, 6.88775427334829457209888409865, 7.68735720796807022216730138863, 8.567175713681758645678410854423, 9.262937693822999194518174337490, 10.62651742082049398720953552246