L(s) = 1 | + (0.5 − 0.866i)2-s − 1.73i·3-s + (−0.499 − 0.866i)4-s + (3 + 1.73i)5-s + (−1.49 − 0.866i)6-s − 2·7-s − 0.999·8-s − 2.99·9-s + (3 − 1.73i)10-s − 1.73i·11-s + (−1.49 + 0.866i)12-s + (−3 + 1.73i)13-s + (−1 + 1.73i)14-s + (2.99 − 5.19i)15-s + (−0.5 + 0.866i)16-s + (6 + 3.46i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s − 0.999i·3-s + (−0.249 − 0.433i)4-s + (1.34 + 0.774i)5-s + (−0.612 − 0.353i)6-s − 0.755·7-s − 0.353·8-s − 0.999·9-s + (0.948 − 0.547i)10-s − 0.522i·11-s + (−0.433 + 0.249i)12-s + (−0.832 + 0.480i)13-s + (−0.267 + 0.462i)14-s + (0.774 − 1.34i)15-s + (−0.125 + 0.216i)16-s + (1.45 + 0.840i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00660 - 0.812287i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00660 - 0.812287i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.73iT \) |
| 19 | \( 1 + (-0.5 - 4.33i)T \) |
good | 5 | \( 1 + (-3 - 1.73i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 1.73iT - 11T^{2} \) |
| 13 | \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-6 - 3.46i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.92iT - 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 - 1.73i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.5 + 2.59i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.5 - 4.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (12 + 6.92i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 5.19iT - 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.5 - 4.33i)T + (48.5 + 84.0i)T^{2} \) |
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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
−13.26631228244715344175451384191, −12.55024267937959604068984170372, −11.44732127937116549735938712841, −10.16640429572740401937683306962, −9.491621912540238649998979429082, −7.75230985102298324916190910150, −6.28164355707810921589486350585, −5.76995324994470245826656273727, −3.24426703404144333301740499515, −1.95196628564662745654120187475,
3.05631423872479306912342985543, 4.98221653212859648678770452758, 5.47662583494971535382675742140, 6.97894429427254294578367153825, 8.708621184955583284032130744406, 9.649821626580960755746413617885, 10.15343836154507203378993261647, 12.02214196569113067546512546877, 12.98303247760488020421352443880, 13.92835200642126841447546786297