L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.5 − 2.59i)5-s + (0.5 + 0.866i)7-s − 0.999·8-s + 3·10-s + (−3 − 5.19i)11-s + (−0.5 + 0.866i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 3·17-s + 2·19-s + (1.50 + 2.59i)20-s + (3 − 5.19i)22-s + (−2 − 3.46i)25-s − 0.999·26-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.670 − 1.16i)5-s + (0.188 + 0.327i)7-s − 0.353·8-s + 0.948·10-s + (−0.904 − 1.56i)11-s + (−0.138 + 0.240i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s − 0.727·17-s + 0.458·19-s + (0.335 + 0.580i)20-s + (0.639 − 1.10i)22-s + (−0.400 − 0.692i)25-s − 0.196·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.546729953\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.546729953\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.711962847609570933809958708012, −8.330576127552605051562552200408, −7.43881914833250116531329591741, −6.31028082722603212969895373561, −5.62669707282006973526322277630, −5.17625283133247283103941897662, −4.26939362641974464016591380286, −3.11577458653625648403268637582, −1.97412007666216217210581245952, −0.44942685657554804936093753684,
1.66240041016730647752147874086, 2.48316605768371302649721145065, 3.24276691934139067756702467121, 4.44445407002124782653118896156, 5.14030210678678737482809281514, 6.05715653733975771558214886617, 7.10655077801468856013375965359, 7.34811904040223645978385172471, 8.649261794348144660180190534618, 9.610904285049721787433692892374