L(s) = 1 | + (0.707 − 0.707i)2-s + i·3-s − 1.00i·4-s + (−0.864 − 0.864i)5-s + (0.707 + 0.707i)6-s + (−1.70 − 2.02i)7-s + (−0.707 − 0.707i)8-s − 9-s − 1.22·10-s + (−3.50 − 3.50i)11-s + 1.00·12-s + (−3.37 + 1.25i)13-s + (−2.63 − 0.222i)14-s + (0.864 − 0.864i)15-s − 1.00·16-s + 0.322·17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + 0.577i·3-s − 0.500i·4-s + (−0.386 − 0.386i)5-s + (0.288 + 0.288i)6-s + (−0.645 − 0.763i)7-s + (−0.250 − 0.250i)8-s − 0.333·9-s − 0.386·10-s + (−1.05 − 1.05i)11-s + 0.288·12-s + (−0.937 + 0.348i)13-s + (−0.704 − 0.0593i)14-s + (0.223 − 0.223i)15-s − 0.250·16-s + 0.0781·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.259027 - 0.890625i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.259027 - 0.890625i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (1.70 + 2.02i)T \) |
| 13 | \( 1 + (3.37 - 1.25i)T \) |
good | 5 | \( 1 + (0.864 + 0.864i)T + 5iT^{2} \) |
| 11 | \( 1 + (3.50 + 3.50i)T + 11iT^{2} \) |
| 17 | \( 1 - 0.322T + 17T^{2} \) |
| 19 | \( 1 + (-1.77 - 1.77i)T + 19iT^{2} \) |
| 23 | \( 1 + 2.70iT - 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + (1.72 + 1.72i)T + 31iT^{2} \) |
| 37 | \( 1 + (-2.27 - 2.27i)T + 37iT^{2} \) |
| 41 | \( 1 + (6.46 + 6.46i)T + 41iT^{2} \) |
| 43 | \( 1 - 0.393iT - 43T^{2} \) |
| 47 | \( 1 + (-1.41 + 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 + 2.03T + 53T^{2} \) |
| 59 | \( 1 + (-10.7 + 10.7i)T - 59iT^{2} \) |
| 61 | \( 1 - 10.6iT - 61T^{2} \) |
| 67 | \( 1 + (-5.38 + 5.38i)T - 67iT^{2} \) |
| 71 | \( 1 + (0.647 - 0.647i)T - 71iT^{2} \) |
| 73 | \( 1 + (6.85 - 6.85i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.81T + 79T^{2} \) |
| 83 | \( 1 + (6.09 + 6.09i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.68 + 3.68i)T - 89iT^{2} \) |
| 97 | \( 1 + (-5.20 - 5.20i)T + 97iT^{2} \) |
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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
−10.33089230151525733826245676524, −9.972680535404967871684054737871, −8.770034270456834728318194174176, −7.84872102174867764733404050812, −6.65785028239023755436351275584, −5.50705093134351372659142881774, −4.61237712227861757295648362818, −3.66235873433317306634373103545, −2.66342208285634907464669039021, −0.42778942485356751555928963875,
2.39022237667538617196839176630, 3.24236346308214543049731319041, 4.85439301430403810092282075615, 5.59326835897816421346362394998, 6.79793550665265457207258344778, 7.38366196788047289084139053680, 8.170308548758432511811046682471, 9.374598842861069691737622927510, 10.21191885449360367748054244494, 11.45750146304296284526519066668