| L(s) = 1 | + (1.35 − 0.399i)2-s + (−0.663 + 1.60i)3-s + (1.68 − 1.08i)4-s + (1.22 − 0.624i)5-s + (−0.260 + 2.43i)6-s + (−1.33 + 0.820i)7-s + (1.84 − 2.14i)8-s + (−0.00546 − 0.00546i)9-s + (1.41 − 1.33i)10-s + (−0.0312 + 0.396i)11-s + (0.621 + 3.41i)12-s + (−1.80 + 0.434i)13-s + (−1.48 + 1.64i)14-s + (0.187 + 2.37i)15-s + (1.64 − 3.64i)16-s + (−3.70 − 4.33i)17-s + ⋯ |
| L(s) = 1 | + (0.959 − 0.282i)2-s + (−0.383 + 0.925i)3-s + (0.840 − 0.542i)4-s + (0.548 − 0.279i)5-s + (−0.106 + 0.995i)6-s + (−0.505 + 0.310i)7-s + (0.652 − 0.757i)8-s + (−0.00182 − 0.00182i)9-s + (0.446 − 0.422i)10-s + (−0.00942 + 0.119i)11-s + (0.179 + 0.985i)12-s + (−0.501 + 0.120i)13-s + (−0.397 + 0.440i)14-s + (0.0483 + 0.614i)15-s + (0.412 − 0.911i)16-s + (−0.898 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.77220 + 0.173678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77220 + 0.173678i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.35 + 0.399i)T \) |
| 41 | \( 1 + (-0.0405 - 6.40i)T \) |
| good | 3 | \( 1 + (0.663 - 1.60i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-1.22 + 0.624i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (1.33 - 0.820i)T + (3.17 - 6.23i)T^{2} \) |
| 11 | \( 1 + (0.0312 - 0.396i)T + (-10.8 - 1.72i)T^{2} \) |
| 13 | \( 1 + (1.80 - 0.434i)T + (11.5 - 5.90i)T^{2} \) |
| 17 | \( 1 + (3.70 + 4.33i)T + (-2.65 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.0591 + 0.246i)T + (-16.9 - 8.62i)T^{2} \) |
| 23 | \( 1 + (0.527 - 0.383i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.35 + 1.58i)T + (-4.53 - 28.6i)T^{2} \) |
| 31 | \( 1 + (3.22 + 9.93i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.115 - 0.355i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (-2.48 - 0.394i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (2.26 + 1.38i)T + (21.3 + 41.8i)T^{2} \) |
| 53 | \( 1 + (-9.34 - 7.98i)T + (8.29 + 52.3i)T^{2} \) |
| 59 | \( 1 + (-6.80 - 9.36i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.04 + 0.798i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (0.540 - 0.0425i)T + (66.1 - 10.4i)T^{2} \) |
| 71 | \( 1 + (-10.9 - 0.862i)T + (70.1 + 11.1i)T^{2} \) |
| 73 | \( 1 + (-3.17 + 3.17i)T - 73iT^{2} \) |
| 79 | \( 1 + (-0.650 - 0.269i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 8.01iT - 83T^{2} \) |
| 89 | \( 1 + (2.54 + 4.15i)T + (-40.4 + 79.2i)T^{2} \) |
| 97 | \( 1 + (18.1 - 1.42i)T + (95.8 - 15.1i)T^{2} \) |
| 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
−13.03785077662912851104848221124, −11.82145139304432193660856773969, −11.04017779493472767063215728808, −9.873797233813360250144042587287, −9.387089712484390346765219145513, −7.32966913921634619746526491847, −6.01295187199331707688054521066, −5.08103168337479920352064950521, −4.10598740529432210978909155626, −2.42499185770852755192802134540,
2.09179816035757114977364122697, 3.78907659202945720444090870888, 5.42019608464757829438243303890, 6.55581362252839855401113602572, 6.97517057825654402183895548888, 8.362181427630125112986366430181, 10.05730409620237874072018326984, 11.06229048269055488875153227563, 12.28720521246375140173608588995, 12.80084802664429276180277363896
