| L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (−0.939 + 0.342i)6-s + (0.716 − 1.24i)7-s + (−0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s + (0.766 + 0.642i)10-s + (−2.65 − 4.60i)11-s + (−0.499 + 0.866i)12-s + (−1.63 + 0.593i)13-s + (−0.248 − 1.41i)14-s + (0.173 − 0.984i)15-s + (−0.939 − 0.342i)16-s + (4.18 − 3.51i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.542 − 0.197i)3-s + (0.0868 − 0.492i)4-s + (0.0776 + 0.440i)5-s + (−0.383 + 0.139i)6-s + (0.270 − 0.469i)7-s + (−0.176 − 0.306i)8-s + (0.255 + 0.214i)9-s + (0.242 + 0.203i)10-s + (−0.801 − 1.38i)11-s + (−0.144 + 0.250i)12-s + (−0.452 + 0.164i)13-s + (−0.0665 − 0.377i)14-s + (0.0448 − 0.254i)15-s + (−0.234 − 0.0855i)16-s + (1.01 − 0.851i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.814686 - 1.22699i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.814686 - 1.22699i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.939 + 0.342i)T \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (-2.91 + 3.24i)T \) |
| good | 7 | \( 1 + (-0.716 + 1.24i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.65 + 4.60i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.63 - 0.593i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-4.18 + 3.51i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.615 + 3.49i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.832 + 0.698i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (3.02 - 5.23i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.140T + 37T^{2} \) |
| 41 | \( 1 + (2.35 + 0.855i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.98 + 11.2i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (1.38 + 1.16i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.911 + 5.16i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.0353 + 0.0296i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (2.54 - 14.4i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.25 - 6.93i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.32 - 7.50i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-6.93 - 2.52i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-12.3 - 4.50i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.740 + 1.28i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.953 + 0.346i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-4.26 + 3.58i)T + (16.8 - 95.5i)T^{2} \) |
| 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
−10.64412364401730070759846922906, −9.997425320567457108742259447167, −8.749430514728033709732024670388, −7.54054181774753266918801922315, −6.80784725910355403356947645733, −5.57399978025887903359153500301, −5.04074346438027979771263043946, −3.58696025278359507269854467164, −2.58015426970805320342996824416, −0.76654736684277777385726468275,
1.89177949274635119433662400265, 3.52397993578843619420730191151, 4.80886556849813050192931914878, 5.32796915394243129350131939732, 6.22803191984465915197311285361, 7.57222927357824420607983490722, 7.971974898213312344442183523714, 9.451148035915201069731690487342, 10.01712409768459005439319081080, 11.14338840517714121223439532791
