| L(s) = 1 | + (−0.104 + 0.994i)2-s + (0.713 − 1.57i)3-s + (−0.978 − 0.207i)4-s + (−1.25 + 2.81i)5-s + (1.49 + 0.874i)6-s + (1.28 + 2.31i)7-s + (0.309 − 0.951i)8-s + (−1.98 − 2.25i)9-s + (−2.66 − 1.54i)10-s + (−3.25 + 0.633i)11-s + (−1.02 + 1.39i)12-s + (−1.85 + 2.55i)13-s + (−2.43 + 1.03i)14-s + (3.54 + 3.98i)15-s + (0.913 + 0.406i)16-s + (0.684 + 6.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (0.412 − 0.911i)3-s + (−0.489 − 0.103i)4-s + (−0.560 + 1.25i)5-s + (0.610 + 0.357i)6-s + (0.485 + 0.874i)7-s + (0.109 − 0.336i)8-s + (−0.660 − 0.750i)9-s + (−0.844 − 0.487i)10-s + (−0.981 + 0.190i)11-s + (−0.296 + 0.402i)12-s + (−0.514 + 0.708i)13-s + (−0.650 + 0.276i)14-s + (0.916 + 1.02i)15-s + (0.228 + 0.101i)16-s + (0.165 + 1.57i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.442857 + 0.909507i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.442857 + 0.909507i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (-0.713 + 1.57i)T \) |
| 7 | \( 1 + (-1.28 - 2.31i)T \) |
| 11 | \( 1 + (3.25 - 0.633i)T \) |
| good | 5 | \( 1 + (1.25 - 2.81i)T + (-3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (1.85 - 2.55i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.684 - 6.50i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (0.878 + 4.13i)T + (-17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (0.815 - 0.470i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.13 - 6.57i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.76 + 3.01i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (0.820 - 0.911i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (0.788 - 2.42i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 1.24iT - 43T^{2} \) |
| 47 | \( 1 + (-2.14 - 10.0i)T + (-42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (2.44 + 5.50i)T + (-35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-2.34 + 11.0i)T + (-53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-3.62 + 8.14i)T + (-40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (3.14 - 5.44i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.06 + 8.34i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.0996 - 0.468i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-16.5 - 1.74i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-9.93 + 7.21i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-5.63 + 3.25i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.13 - 5.18i)T + (29.9 + 92.2i)T^{2} \) |
| 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
−11.40718873150579527555397827270, −10.54829381422755348060030681395, −9.303315458518074634400965595781, −8.244829421245639427726553001728, −7.78188643516610373920108748290, −6.78773608603408678145038711551, −6.16560683833176603209487427969, −4.81166354882047373561239137302, −3.22625917284651417661981462551, −2.14570224817648487176228194245,
0.60273140959587242929098453152, 2.62842514742403269017779823766, 3.90657168847661497954311320345, 4.76113518829383396673297164034, 5.34476930972354901751597704155, 7.65063859109762029836828730271, 8.132773082656789596160169342229, 8.988897401540100300953811966653, 10.11264822868568890696144467169, 10.43085294514513490614386515760
