L(s) = 1 | + (−1.60 + 2.77i)5-s + (0.555 + 2.58i)7-s + (4.13 − 2.38i)11-s + (−0.861 − 0.497i)13-s + 6.51·17-s + 5.38i·19-s + (6.56 + 3.79i)23-s + (−2.63 − 4.57i)25-s + (2.93 − 1.69i)29-s + (−3.51 − 2.02i)31-s + (−8.07 − 2.60i)35-s + 6.29·37-s + (3.48 − 6.04i)41-s + (−3.81 − 6.60i)43-s + (3.78 + 6.54i)47-s + ⋯ |
L(s) = 1 | + (−0.716 + 1.24i)5-s + (0.209 + 0.977i)7-s + (1.24 − 0.720i)11-s + (−0.239 − 0.137i)13-s + 1.57·17-s + 1.23i·19-s + (1.36 + 0.790i)23-s + (−0.527 − 0.914i)25-s + (0.545 − 0.314i)29-s + (−0.630 − 0.364i)31-s + (−1.36 − 0.440i)35-s + 1.03·37-s + (0.544 − 0.943i)41-s + (−0.581 − 1.00i)43-s + (0.551 + 0.955i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0329 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0329 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.910140801\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.910140801\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.555 - 2.58i)T \) |
good | 5 | \( 1 + (1.60 - 2.77i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.13 + 2.38i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.861 + 0.497i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 6.51T + 17T^{2} \) |
| 19 | \( 1 - 5.38iT - 19T^{2} \) |
| 23 | \( 1 + (-6.56 - 3.79i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.93 + 1.69i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.51 + 2.02i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.29T + 37T^{2} \) |
| 41 | \( 1 + (-3.48 + 6.04i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.81 + 6.60i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.78 - 6.54i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 7.77iT - 53T^{2} \) |
| 59 | \( 1 + (2.25 - 3.90i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.26 + 3.03i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.493 - 0.854i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.20iT - 71T^{2} \) |
| 73 | \( 1 + 2.63iT - 73T^{2} \) |
| 79 | \( 1 + (-7.96 - 13.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.49 + 7.77i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 4.42T + 89T^{2} \) |
| 97 | \( 1 + (3.13 - 1.81i)T + (48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.913447010303153305463146421031, −7.997580938173038002764938716527, −7.51268922985648567104042487576, −6.63181058649371230726613628759, −5.90510212100480592358308625491, −5.26142632920320436826987515171, −3.82266986971222491058636709166, −3.42916267081378726979633807639, −2.52237375444111247059971817296, −1.17704959527257574718809504436,
0.78203282807360366522355016134, 1.34714676546090019263244830575, 3.00907287507089266583553483937, 4.02474291141208208232032701232, 4.62621057434683008659159193360, 5.11247048078503504549005948649, 6.44386883776161111995638283277, 7.17085989163124385436305386331, 7.74007877653147365530458387760, 8.577291788975010783831615775496