| L(s) = 1 | + (−1.62 − 0.937i)5-s + (0.0301 − 0.0174i)7-s + (−2.78 − 4.83i)11-s + (0.468 − 0.811i)13-s − 1.47i·17-s − 0.534i·19-s + (2.06 − 3.58i)23-s + (−0.741 − 1.28i)25-s + (−7.52 + 4.34i)29-s + (5.02 + 2.90i)31-s − 0.0654·35-s + 3.56·37-s + (0.0657 + 0.0379i)41-s + (−6.31 + 3.64i)43-s + (4.70 + 8.14i)47-s + ⋯ |
| L(s) = 1 | + (−0.726 − 0.419i)5-s + (0.0114 − 0.00658i)7-s + (−0.841 − 1.45i)11-s + (0.129 − 0.225i)13-s − 0.357i·17-s − 0.122i·19-s + (0.431 − 0.746i)23-s + (−0.148 − 0.256i)25-s + (−1.39 + 0.806i)29-s + (0.902 + 0.521i)31-s − 0.0110·35-s + 0.586·37-s + (0.0102 + 0.00593i)41-s + (−0.963 + 0.556i)43-s + (0.685 + 1.18i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1589321377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1589321377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.62 + 0.937i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.0301 + 0.0174i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.78 + 4.83i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.468 + 0.811i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.47iT - 17T^{2} \) |
| 19 | \( 1 + 0.534iT - 19T^{2} \) |
| 23 | \( 1 + (-2.06 + 3.58i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (7.52 - 4.34i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.02 - 2.90i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.56T + 37T^{2} \) |
| 41 | \( 1 + (-0.0657 - 0.0379i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.31 - 3.64i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.70 - 8.14i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 11.4iT - 53T^{2} \) |
| 59 | \( 1 + (-4.04 + 7.01i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.91 - 11.9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.53 + 4.34i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.66T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + (3.57 - 2.06i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.414 - 0.717i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 6.55iT - 89T^{2} \) |
| 97 | \( 1 + (0.909 + 1.57i)T + (-48.5 + 84.0i)T^{2} \) |
| 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
−8.223982031708032464504767751927, −7.59597582256564296831183660025, −6.66197100946923587321825940081, −5.81399471160597589810161047494, −5.10001637943540854345855255190, −4.29589351306675691164930626428, −3.32814738927024634537060212704, −2.66380666111618120053454666709, −1.09547417158466248303652785399, −0.05255174411956745581677822938,
1.69871178845133165407037415000, 2.60777774080102443986475911503, 3.67925373832242842429914841259, 4.32672321220710102195654741975, 5.22129196080096746139504711236, 6.02479423259550821665301516227, 7.13471899969121979002741635534, 7.42128582677446024012741341285, 8.121257345535896960937868277216, 9.019531621099935525899644875271
