L(s) = 1 | + (0.866 − 0.5i)2-s + (1.13 + 1.30i)3-s + (0.499 − 0.866i)4-s + 5-s + (1.63 + 0.560i)6-s + (2.26 − 1.36i)7-s − 0.999i·8-s + (−0.403 + 2.97i)9-s + (0.866 − 0.5i)10-s + 3.46i·11-s + (1.69 − 0.334i)12-s + (−0.584 + 0.337i)13-s + (1.28 − 2.31i)14-s + (1.13 + 1.30i)15-s + (−0.5 − 0.866i)16-s + (−2.09 − 3.62i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.657 + 0.753i)3-s + (0.249 − 0.433i)4-s + 0.447·5-s + (0.669 + 0.228i)6-s + (0.856 − 0.515i)7-s − 0.353i·8-s + (−0.134 + 0.990i)9-s + (0.273 − 0.158i)10-s + 1.04i·11-s + (0.490 − 0.0965i)12-s + (−0.162 + 0.0936i)13-s + (0.342 − 0.618i)14-s + (0.294 + 0.336i)15-s + (−0.125 − 0.216i)16-s + (−0.507 − 0.878i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.89665 + 0.152855i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.89665 + 0.152855i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (-1.13 - 1.30i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.26 + 1.36i)T \) |
good | 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + (0.584 - 0.337i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.09 + 3.62i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.18 - 0.683i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 2.10iT - 23T^{2} \) |
| 29 | \( 1 + (4.77 + 2.75i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.25 + 0.724i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.59 + 7.95i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.79 - 8.30i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.38 - 2.39i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.50 - 4.33i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.77 - 3.90i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.15 - 3.73i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.15 - 2.97i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.90 - 3.29i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 15.0iT - 71T^{2} \) |
| 73 | \( 1 + (-2.54 + 1.46i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.65 + 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.69 - 2.93i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.11 - 5.39i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.750 + 0.433i)T + (48.5 + 84.0i)T^{2} \) |
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
−10.72355526965529448760920080003, −9.674841099697811535178294763567, −9.269640536685507194506576976761, −7.894902249374616429247548400699, −7.20267991789876238165876758151, −5.77142834294397259470767541177, −4.65818170815820632703554668090, −4.28963872195869567476454162214, −2.81573085965642035787841271506, −1.81747733663075277896345412573,
1.62109101569272911962749944138, 2.75096740448407938603786580239, 3.88280073879582782237845002930, 5.30348695964306263131201697293, 6.02003703502116096579612072447, 6.98343485094473986201787662515, 7.996594959654121877284011530792, 8.578008194342694605387701831706, 9.404019414510098753135847972098, 10.84651811084952600489697210946