| L(s) = 1 | + (−0.978 − 0.207i)2-s + (1.67 − 0.446i)3-s + (0.913 + 0.406i)4-s + (−2.23 + 0.0195i)5-s + (−1.72 + 0.0885i)6-s + (2.28 − 3.95i)7-s + (−0.809 − 0.587i)8-s + (2.60 − 1.49i)9-s + (2.19 + 0.445i)10-s + (−3.20 − 0.681i)11-s + (1.71 + 0.272i)12-s + (−0.940 + 0.199i)13-s + (−3.05 + 3.39i)14-s + (−3.73 + 1.03i)15-s + (0.669 + 0.743i)16-s + (1.23 + 0.895i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (0.966 − 0.257i)3-s + (0.456 + 0.203i)4-s + (−0.999 + 0.00872i)5-s + (−0.706 + 0.0361i)6-s + (0.863 − 1.49i)7-s + (−0.286 − 0.207i)8-s + (0.867 − 0.497i)9-s + (0.692 + 0.140i)10-s + (−0.966 − 0.205i)11-s + (0.493 + 0.0788i)12-s + (−0.260 + 0.0554i)13-s + (−0.817 + 0.907i)14-s + (−0.963 + 0.266i)15-s + (0.167 + 0.185i)16-s + (0.298 + 0.217i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0964 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0964 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.772893 - 0.851440i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.772893 - 0.851440i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 3 | \( 1 + (-1.67 + 0.446i)T \) |
| 5 | \( 1 + (2.23 - 0.0195i)T \) |
| good | 7 | \( 1 + (-2.28 + 3.95i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.20 + 0.681i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.940 - 0.199i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-1.23 - 0.895i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (4.38 + 3.18i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.90 + 2.11i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.0431 + 0.410i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.354 + 3.36i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.111 - 0.342i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-9.84 + 2.09i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (4.32 - 7.49i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.0278 + 0.265i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-8.22 + 5.97i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (12.6 - 2.67i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-12.5 - 2.66i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (0.0195 - 0.185i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-2.90 + 2.10i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.34 - 13.3i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.548 + 5.21i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (10.2 - 4.58i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (0.871 + 2.68i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.61 - 15.3i)T + (-94.8 + 20.1i)T^{2} \) |
| 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
−10.76855354047682788823495539536, −10.05023795496983025818620759402, −8.772036927008438148088675305692, −8.019906710536406341912694646674, −7.56131571000562158489288749012, −6.77063596922891602017429059214, −4.66470061208447045789009716498, −3.81482237559264195364532624084, −2.50494629515977153449316430669, −0.823833558612241761823844728789,
2.01353390000654324886642015538, 3.04502011943500142998851317814, 4.55808776969402634538515335383, 5.56406477829827663249039989589, 7.19855169653168818985703649337, 8.014469680015317524716713821229, 8.475175330631246621255675127045, 9.244677354618496961679141479611, 10.35625236772803723378047711372, 11.16258212492063326376345972548
