L(s) = 1 | + (−1.02 + 0.593i)2-s + (−0.295 + 0.511i)4-s + (−1.44 + 1.71i)5-s + (−1.75 − 1.01i)7-s − 3.07i·8-s + (0.465 − 2.61i)10-s + (1.94 + 3.36i)11-s + (−2.96 + 2.05i)13-s + 2.40·14-s + (1.23 + 2.14i)16-s + (−4.71 − 2.72i)17-s + (2.94 − 5.09i)19-s + (−0.449 − 1.24i)20-s + (−3.99 − 2.30i)22-s + (−0.298 + 0.172i)23-s + ⋯ |
L(s) = 1 | + (−0.727 + 0.419i)2-s + (−0.147 + 0.255i)4-s + (−0.644 + 0.764i)5-s + (−0.664 − 0.383i)7-s − 1.08i·8-s + (0.147 − 0.826i)10-s + (0.585 + 1.01i)11-s + (−0.821 + 0.570i)13-s + 0.644·14-s + (0.308 + 0.535i)16-s + (−1.14 − 0.660i)17-s + (0.674 − 1.16i)19-s + (−0.100 − 0.277i)20-s + (−0.850 − 0.491i)22-s + (−0.0623 + 0.0359i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.145565 - 0.122106i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.145565 - 0.122106i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.44 - 1.71i)T \) |
| 13 | \( 1 + (2.96 - 2.05i)T \) |
good | 2 | \( 1 + (1.02 - 0.593i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (1.75 + 1.01i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.94 - 3.36i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (4.71 + 2.72i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.94 + 5.09i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.298 - 0.172i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.18T + 31T^{2} \) |
| 37 | \( 1 + (-4.71 + 2.72i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0902 - 0.156i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.15 + 0.669i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 12.2iT - 47T^{2} \) |
| 53 | \( 1 - 2.42iT - 53T^{2} \) |
| 59 | \( 1 + (-3.53 + 6.11i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.38 - 5.85i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.81 - 2.20i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.940 - 1.62i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 8.86iT - 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 7.83iT - 83T^{2} \) |
| 89 | \( 1 + (6.12 + 10.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.02 - 2.90i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.18158436071831742755429937657, −9.549475792515759457647917734399, −8.865552497179629927015860626798, −7.58995597284769273858895257517, −7.02339732120502868875506396384, −6.62577727030182663289615531877, −4.65459375893594842925235050424, −3.85629849993693549631355554422, −2.57876203079263162072378718853, −0.14730469439585271047318327932,
1.31273820435065317068897950309, 2.96133333651627281707083712305, 4.25527101927818831515175506208, 5.42481657544912142339621726342, 6.24702888003597683337266505702, 7.74653138686344573099104816666, 8.468576628035327469725888563640, 9.190605453917871689989413397611, 9.847724990451877032254146097988, 10.86777233961915383488501823348