L(s) = 1 | + (1.93 − 1.11i)2-s + (1.5 − 2.59i)4-s + (−1.93 + 3.35i)5-s + (2.5 + 0.866i)7-s − 2.23i·8-s + 8.66i·10-s + (1.93 − 1.11i)11-s + (3 + 1.73i)13-s + (5.80 − 1.11i)14-s + (0.499 + 0.866i)16-s − 5.19i·19-s + (5.80 + 10.0i)20-s + (2.5 − 4.33i)22-s + (−1.93 − 1.11i)23-s + (−5.00 − 8.66i)25-s + 7.74·26-s + ⋯ |
L(s) = 1 | + (1.36 − 0.790i)2-s + (0.750 − 1.29i)4-s + (−0.866 + 1.50i)5-s + (0.944 + 0.327i)7-s − 0.790i·8-s + 2.73i·10-s + (0.583 − 0.337i)11-s + (0.832 + 0.480i)13-s + (1.55 − 0.298i)14-s + (0.124 + 0.216i)16-s − 1.19i·19-s + (1.29 + 2.25i)20-s + (0.533 − 0.923i)22-s + (−0.403 − 0.233i)23-s + (−1.00 − 1.73i)25-s + 1.51·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.93868 - 0.234030i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.93868 - 0.234030i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1.93 + 1.11i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.93 - 3.35i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.93 + 1.11i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3 - 1.73i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.19iT - 19T^{2} \) |
| 23 | \( 1 + (1.93 + 1.11i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.87 - 2.23i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (1.93 - 3.35i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.87 + 6.70i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 8.94iT - 53T^{2} \) |
| 59 | \( 1 + (-3.87 + 6.70i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.1iT - 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.87 + 6.70i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + (-12 + 6.92i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.06659487393032292960853797049, −10.54985750948002223611926424224, −8.990959417125780112761976727609, −7.927709407128270379635016921442, −6.83567068908960919764116912653, −6.03731989250275003562667513781, −4.79147558419075862325611196995, −3.86496772589655833612195259618, −3.11389435560840924760071421220, −1.94106768991506357502833670448,
1.35352997901084349026376865349, 3.77305798286006032997579131066, 4.16721308387248423838542376528, 5.15448782288626654496235070064, 5.82069851885546312400547872077, 7.15588130562237895904015758190, 8.017218441928453461422134723055, 8.509476020893268340061482849346, 9.813075513427967085418928106491, 11.27087692919219765790822126362