| L(s) = 1 | + (−0.922 + 1.07i)2-s + (0.819 − 0.573i)3-s + (−0.299 − 1.97i)4-s + (−3.12 + 0.273i)5-s + (−0.140 + 1.40i)6-s + (−0.765 − 1.32i)7-s + (2.39 + 1.50i)8-s + (0.342 − 0.939i)9-s + (2.58 − 3.60i)10-s + (4.47 − 1.19i)11-s + (−1.37 − 1.44i)12-s + (2.92 + 2.04i)13-s + (2.12 + 0.401i)14-s + (−2.40 + 2.01i)15-s + (−3.82 + 1.18i)16-s + (−3.10 + 1.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.652 + 0.758i)2-s + (0.472 − 0.331i)3-s + (−0.149 − 0.988i)4-s + (−1.39 + 0.122i)5-s + (−0.0572 + 0.574i)6-s + (−0.289 − 0.501i)7-s + (0.847 + 0.531i)8-s + (0.114 − 0.313i)9-s + (0.818 − 1.13i)10-s + (1.34 − 0.361i)11-s + (−0.398 − 0.418i)12-s + (0.810 + 0.567i)13-s + (0.568 + 0.107i)14-s + (−0.620 + 0.520i)15-s + (−0.955 + 0.296i)16-s + (−0.753 + 0.274i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.228828 - 0.360662i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228828 - 0.360662i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.922 - 1.07i)T \) |
| 3 | \( 1 + (-0.819 + 0.573i)T \) |
| 19 | \( 1 + (2.98 + 3.17i)T \) |
| good | 5 | \( 1 + (3.12 - 0.273i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (0.765 + 1.32i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.47 + 1.19i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-2.92 - 2.04i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (3.10 - 1.13i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (3.02 - 2.53i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (6.18 - 2.88i)T + (18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (4.89 + 8.47i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.45 + 4.45i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.872 - 4.94i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.665 + 7.61i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-4.37 + 12.0i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.04 - 0.0912i)T + (52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (1.66 - 3.57i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (8.33 + 0.729i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (3.13 + 6.73i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (7.49 - 8.93i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.515 + 0.0908i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (2.49 - 14.1i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.96 - 0.795i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-0.119 - 0.677i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-1.75 - 4.82i)T + (-74.3 + 62.3i)T^{2} \) |
| 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
−9.389788314825890679747730862626, −8.844294347258953946545348927768, −8.188763852132202936993496598139, −7.13927566626407120121383121351, −6.87874766450322288481022039404, −5.80662149808266651368048953780, −4.02970224708947855302592108118, −3.88630477214896821354597007114, −1.78950298670747507250786687090, −0.24395347947011471220345487902,
1.62187788615078994829918847806, 3.10043387628592981087648895367, 3.88174723495637267523031807627, 4.48023637477941226525829611477, 6.25092161683738454280768544598, 7.34118010171602226649894822787, 8.111035111061987230475658943887, 8.862534362459690040234201302648, 9.269400587355343712922471291913, 10.47568802842553019896760552237
