L(s) = 1 | + (−0.707 − 1.22i)3-s + (0.500 − 0.866i)9-s + (−1 − 1.73i)11-s + 2.82·13-s + (−2.12 − 3.67i)17-s + (2.12 − 3.67i)19-s + (−4 + 6.92i)23-s + (2.5 + 4.33i)25-s − 5.65·27-s + 6·29-s + (−4.24 − 7.34i)31-s + (−1.41 + 2.44i)33-s + (1 − 1.73i)37-s + (−2.00 − 3.46i)39-s − 4.24·41-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.707i)3-s + (0.166 − 0.288i)9-s + (−0.301 − 0.522i)11-s + 0.784·13-s + (−0.514 − 0.891i)17-s + (0.486 − 0.842i)19-s + (−0.834 + 1.44i)23-s + (0.5 + 0.866i)25-s − 1.08·27-s + 1.11·29-s + (−0.762 − 1.31i)31-s + (−0.246 + 0.426i)33-s + (0.164 − 0.284i)37-s + (−0.320 − 0.554i)39-s − 0.662·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.108735366\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.108735366\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.707 + 1.22i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + (2.12 + 3.67i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.12 + 3.67i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (4.24 + 7.34i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.36 + 11.0i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.82 + 4.89i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + (-0.707 - 1.22i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + (2.12 - 3.67i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−9.250291068844174516040138689066, −8.207547191631340638646196799555, −7.40461836570732269526638749636, −6.74938163587396926776535545588, −5.90925197614927873912122679531, −5.19084452289736276253001716497, −3.96305907775527282941295261681, −3.00064725385592036977044219661, −1.65487196298940704089874676797, −0.46977527012987877848357091576,
1.53054612525723678399969761225, 2.81853297801407290945965453442, 4.15141449911643821681643213272, 4.54460729804461556703721270043, 5.66098856986975973444086923494, 6.33790430444952708972451610287, 7.34215358549945915458458301182, 8.314865577479108823500476304584, 8.848078399847061168812954031477, 10.11340448176609691386900599733