L(s) = 1 | + (0.866 − 1.5i)5-s + (−1.62 + 2.09i)7-s + (3.67 − 2.12i)11-s + (−2.12 − 1.22i)13-s − 1.73·17-s − 2.44i·19-s + (−7.34 − 4.24i)23-s + (1 + 1.73i)25-s + (3.67 − 2.12i)29-s + (−6.36 − 3.67i)31-s + (1.73 + 4.24i)35-s + 37-s + (−0.866 + 1.5i)41-s + (−3.5 − 6.06i)43-s + (6.06 + 10.5i)47-s + ⋯ |
L(s) = 1 | + (0.387 − 0.670i)5-s + (−0.612 + 0.790i)7-s + (1.10 − 0.639i)11-s + (−0.588 − 0.339i)13-s − 0.420·17-s − 0.561i·19-s + (−1.53 − 0.884i)23-s + (0.200 + 0.346i)25-s + (0.682 − 0.393i)29-s + (−1.14 − 0.659i)31-s + (0.292 + 0.717i)35-s + 0.164·37-s + (−0.135 + 0.234i)41-s + (−0.533 − 0.924i)43-s + (0.884 + 1.53i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.130003402\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.130003402\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.62 - 2.09i)T \) |
good | 5 | \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.67 + 2.12i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.12 + 1.22i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 23 | \( 1 + (7.34 + 4.24i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.67 + 2.12i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (6.36 + 3.67i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + (0.866 - 1.5i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.5 + 6.06i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.06 - 10.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (-4.33 + 7.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.12 - 1.22i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.06 + 10.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + (-2.12 + 1.22i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.849950104536672211182345576014, −8.248854209300262648265848710330, −7.13015104713568688117955416095, −6.23908994311134362331404556227, −5.77383138078713442693425159169, −4.80127899311515555304139166859, −3.90137551496666585400934476229, −2.81621323096548305166420930373, −1.84333491368649619061893591297, −0.37670627136664649667685616976,
1.46078960532197698852893184609, 2.49359136042701177376134405642, 3.70327274258445749065813927654, 4.19595520815506785794525403622, 5.41069507527942382300581682412, 6.41996656462520248859563373881, 6.86695545534883471915257983278, 7.49360201248938750367203733486, 8.571358187140873774766008210806, 9.486869579936277668472782189174