| L(s) = 1 | + (−2.65 + 0.468i)2-s + (2.97 + 0.359i)3-s + (3.09 − 1.12i)4-s + (0.583 + 0.695i)5-s + (−8.08 + 0.439i)6-s + (−5.66 − 9.81i)7-s + (1.65 − 0.954i)8-s + (8.74 + 2.14i)9-s + (−1.87 − 1.57i)10-s − 5.05i·11-s + (9.62 − 2.24i)12-s + (−15.5 − 13.0i)13-s + (19.6 + 23.4i)14-s + (1.48 + 2.28i)15-s + (−14.0 + 11.7i)16-s + (11.6 + 13.8i)17-s + ⋯ |
| L(s) = 1 | + (−1.32 + 0.234i)2-s + (0.992 + 0.119i)3-s + (0.773 − 0.281i)4-s + (0.116 + 0.139i)5-s + (−1.34 + 0.0732i)6-s + (−0.809 − 1.40i)7-s + (0.206 − 0.119i)8-s + (0.971 + 0.238i)9-s + (−0.187 − 0.157i)10-s − 0.459i·11-s + (0.801 − 0.186i)12-s + (−1.19 − 1.00i)13-s + (1.40 + 1.67i)14-s + (0.0992 + 0.152i)15-s + (−0.877 + 0.736i)16-s + (0.682 + 0.813i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.757658 - 0.428362i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.757658 - 0.428362i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.97 - 0.359i)T \) |
| 19 | \( 1 + (-13.2 + 13.5i)T \) |
| good | 2 | \( 1 + (2.65 - 0.468i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-0.583 - 0.695i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (5.66 + 9.81i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + 5.05iT - 121T^{2} \) |
| 13 | \( 1 + (15.5 + 13.0i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-11.6 - 13.8i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (3.99 + 10.9i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (9.83 + 27.0i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 - 57.8T + 961T^{2} \) |
| 37 | \( 1 + 34.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-38.6 + 6.82i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (21.1 + 7.70i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-26.5 - 72.9i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (82.3 + 14.5i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (16.3 - 44.9i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-8.35 - 7.01i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-4.49 + 25.4i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (48.8 - 8.62i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-63.2 - 23.0i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (49.8 - 41.8i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-70.1 + 40.4i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-32.6 - 89.7i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (9.06 + 51.3i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.53672472826622400380337289966, −10.60116550280618980392879363093, −10.11117207965851573637275266781, −9.463351693890525871815870545861, −8.127055867130420734885829068753, −7.59984965153391638333692277765, −6.56709846237182970080910742707, −4.33144909528639427422396512074, −2.90654121609370817914916734210, −0.75232490382319845466463812780,
1.81259434217032970044211820825, 2.98512713831309589379566150917, 5.06393300554002830361560439221, 6.89610387834665921022542112796, 7.80943190605803798771234007781, 8.966085855272574233448423255346, 9.508943222713624761027913722785, 9.993942114799276954542195784546, 11.80878803980477085016422960976, 12.39297734442214836376303328862
