| L(s) = 1 | + (−0.766 − 0.642i)2-s + (−1.07 + 1.35i)3-s + (0.173 + 0.984i)4-s + (−0.984 − 0.173i)5-s + (1.69 − 0.345i)6-s + (−0.267 − 0.464i)7-s + (0.500 − 0.866i)8-s + (−0.673 − 2.92i)9-s + (0.642 + 0.766i)10-s + (−0.233 − 0.134i)11-s + (−1.52 − 0.826i)12-s + (−0.650 + 1.78i)13-s + (−0.0930 + 0.527i)14-s + (1.29 − 1.14i)15-s + (−0.939 + 0.342i)16-s + (0.328 − 0.391i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.622 + 0.782i)3-s + (0.0868 + 0.492i)4-s + (−0.440 − 0.0776i)5-s + (0.692 − 0.140i)6-s + (−0.101 − 0.175i)7-s + (0.176 − 0.306i)8-s + (−0.224 − 0.974i)9-s + (0.203 + 0.242i)10-s + (−0.0703 − 0.0406i)11-s + (−0.439 − 0.238i)12-s + (−0.180 + 0.495i)13-s + (−0.0248 + 0.141i)14-s + (0.334 − 0.296i)15-s + (−0.234 + 0.0855i)16-s + (0.0795 − 0.0948i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.571132 - 0.317327i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.571132 - 0.317327i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.07 - 1.35i)T \) |
| 5 | \( 1 + (0.984 + 0.173i)T \) |
| 19 | \( 1 + (0.783 + 4.28i)T \) |
| good | 7 | \( 1 + (0.267 + 0.464i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.233 + 0.134i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.650 - 1.78i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.328 + 0.391i)T + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.229 - 0.0404i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.10 + 3.44i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-6.10 + 3.52i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.16iT - 37T^{2} \) |
| 41 | \( 1 + (-10.9 + 3.97i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.13 - 6.46i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.83 - 6.95i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (0.990 + 5.61i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (4.93 + 4.14i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.85 + 10.5i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (3.68 + 4.39i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.11 - 6.31i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (5.29 - 1.92i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.34 - 6.45i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.54 + 1.46i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.978 + 0.356i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.37 + 1.64i)T + (-16.8 - 95.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.72985379816971130410849655507, −9.694639996707161126406700934814, −9.163424623133210884174723735038, −8.102577393397033333546689613106, −7.04386009714059173210857674627, −6.04214537808418463913580568336, −4.70218922483905166163702903271, −3.98444059872094039195339460227, −2.66293831720887025940321306728, −0.57267346460579843724693412662,
1.16456890517733505717623956011, 2.78098720873341022776788555968, 4.54964578274897742579189475690, 5.66658996513511068414888124440, 6.42456481418980355218545035222, 7.37879955950151914217523309748, 8.047627060182614568074401462343, 8.864382568619523929918412843394, 10.25394820519546334727538408552, 10.65768892261030433680559732124
