L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−2 − 3.46i)5-s + (−2.5 − 0.866i)7-s + 0.999·8-s + (−1.99 + 3.46i)10-s + (−0.5 + 0.866i)11-s − 13-s + (0.500 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (1 − 1.73i)17-s + (−3 − 5.19i)19-s + 3.99·20-s + 0.999·22-s + (−1 − 1.73i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.894 − 1.54i)5-s + (−0.944 − 0.327i)7-s + 0.353·8-s + (−0.632 + 1.09i)10-s + (−0.150 + 0.261i)11-s − 0.277·13-s + (0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.242 − 0.420i)17-s + (−0.688 − 1.19i)19-s + 0.894·20-s + 0.213·22-s + (−0.208 − 0.361i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.5 + 7.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.5 - 12.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−9.201647939686874887845029844992, −8.237682208326296493849964574679, −7.54603717305674134171837986224, −6.62760607304414910483891663431, −5.25034017071322233503876085325, −4.49395480179660569319583333273, −3.75101847876993285462280275100, −2.57653131589788772638924663908, −1.00503494855272045806181227706, 0,
2.29754830802319300278492286719, 3.41286312951143572308018911181, 4.05122305797424573947698127282, 5.66563171531911962519381403483, 6.27751924547485755292347136030, 7.01367227274302486257420815689, 7.72911967490789593129587584884, 8.381601595198172497295150491774, 9.502920509103336239377245719260