L(s) = 1 | + (0.602 + 0.602i)3-s + (−3.63 + 3.43i)5-s + (−6.67 + 6.67i)7-s − 8.27i·9-s − 18.9·11-s + (7.47 + 7.47i)13-s + (−4.25 − 0.123i)15-s + (1.45 − 1.45i)17-s + 24.9i·19-s − 8.04·21-s + (3.74 + 3.74i)23-s + (1.45 − 24.9i)25-s + (10.4 − 10.4i)27-s + 29.8i·29-s − 21.0·31-s + ⋯ |
L(s) = 1 | + (0.200 + 0.200i)3-s + (−0.727 + 0.686i)5-s + (−0.952 + 0.952i)7-s − 0.919i·9-s − 1.72·11-s + (0.575 + 0.575i)13-s + (−0.283 − 0.00825i)15-s + (0.0854 − 0.0854i)17-s + 1.31i·19-s − 0.382·21-s + (0.162 + 0.162i)23-s + (0.0580 − 0.998i)25-s + (0.385 − 0.385i)27-s + 1.02i·29-s − 0.679·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.550i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.189896 + 0.633318i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.189896 + 0.633318i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (3.63 - 3.43i)T \) |
good | 3 | \( 1 + (-0.602 - 0.602i)T + 9iT^{2} \) |
| 7 | \( 1 + (6.67 - 6.67i)T - 49iT^{2} \) |
| 11 | \( 1 + 18.9T + 121T^{2} \) |
| 13 | \( 1 + (-7.47 - 7.47i)T + 169iT^{2} \) |
| 17 | \( 1 + (-1.45 + 1.45i)T - 289iT^{2} \) |
| 19 | \( 1 - 24.9iT - 361T^{2} \) |
| 23 | \( 1 + (-3.74 - 3.74i)T + 529iT^{2} \) |
| 29 | \( 1 - 29.8iT - 841T^{2} \) |
| 31 | \( 1 + 21.0T + 961T^{2} \) |
| 37 | \( 1 + (18.8 - 18.8i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 40.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-37.5 - 37.5i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-48.9 + 48.9i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (33.9 + 33.9i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 20.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 19.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-35.9 + 35.9i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 120.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-85.6 - 85.6i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 75.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (21.5 + 21.5i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 43.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (103. - 103. i)T - 9.40e3iT^{2} \) |
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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
−12.79852891390706282254161848008, −12.21384411407329199376926309450, −11.02121337295222044072309820863, −10.04025497176559789187332270280, −8.995848063267163868105104108464, −7.917074513826016450038285298912, −6.67082842534932850585796771050, −5.60512055503224343030255844687, −3.74557573447888078603450764000, −2.78293226343034603618046192985,
0.38644066909947459034475889058, 2.84308142511606184923531486980, 4.33413058225275128735113779089, 5.56397257145596247494723771659, 7.32331257237832810705211018936, 7.86350707037679531666215996733, 9.058166511784788887665254832189, 10.46627114571008241783463886779, 10.99331312970643579865651908273, 12.65070517727806680837883972786