| L(s) = 1 | + (−0.748 + 0.663i)2-s + (0.298 − 0.432i)3-s + (0.120 − 0.992i)4-s + (−0.874 + 0.215i)5-s + (0.0633 + 0.521i)6-s + (2.93 − 4.24i)7-s + (0.568 + 0.822i)8-s + (0.965 + 2.54i)9-s + (0.511 − 0.741i)10-s + (1.81 + 0.446i)11-s + (−0.393 − 0.348i)12-s + (0.402 − 3.31i)13-s + (0.622 + 5.12i)14-s + (−0.167 + 0.442i)15-s + (−0.970 − 0.239i)16-s + (−0.154 + 1.27i)17-s + ⋯ |
| L(s) = 1 | + (−0.529 + 0.468i)2-s + (0.172 − 0.249i)3-s + (0.0602 − 0.496i)4-s + (−0.391 + 0.0964i)5-s + (0.0258 + 0.213i)6-s + (1.10 − 1.60i)7-s + (0.200 + 0.290i)8-s + (0.321 + 0.848i)9-s + (0.161 − 0.234i)10-s + (0.546 + 0.134i)11-s + (−0.113 − 0.100i)12-s + (0.111 − 0.920i)13-s + (0.166 + 1.36i)14-s + (−0.0433 + 0.114i)15-s + (−0.242 − 0.0598i)16-s + (−0.0374 + 0.308i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.983288 - 0.0681701i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.983288 - 0.0681701i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.748 - 0.663i)T \) |
| 79 | \( 1 + (-8.57 + 2.33i)T \) |
| good | 3 | \( 1 + (-0.298 + 0.432i)T + (-1.06 - 2.80i)T^{2} \) |
| 5 | \( 1 + (0.874 - 0.215i)T + (4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (-2.93 + 4.24i)T + (-2.48 - 6.54i)T^{2} \) |
| 11 | \( 1 + (-1.81 - 0.446i)T + (9.74 + 5.11i)T^{2} \) |
| 13 | \( 1 + (-0.402 + 3.31i)T + (-12.6 - 3.11i)T^{2} \) |
| 17 | \( 1 + (0.154 - 1.27i)T + (-16.5 - 4.06i)T^{2} \) |
| 19 | \( 1 + (-2.26 - 1.18i)T + (10.7 + 15.6i)T^{2} \) |
| 23 | \( 1 + 0.623T + 23T^{2} \) |
| 29 | \( 1 + (-2.28 + 6.02i)T + (-21.7 - 19.2i)T^{2} \) |
| 31 | \( 1 + (7.05 - 6.25i)T + (3.73 - 30.7i)T^{2} \) |
| 37 | \( 1 + (1.71 + 0.899i)T + (21.0 + 30.4i)T^{2} \) |
| 41 | \( 1 + (9.75 - 2.40i)T + (36.3 - 19.0i)T^{2} \) |
| 43 | \( 1 + (-1.22 + 0.302i)T + (38.0 - 19.9i)T^{2} \) |
| 47 | \( 1 + (6.10 - 3.20i)T + (26.6 - 38.6i)T^{2} \) |
| 53 | \( 1 + (-3.36 - 4.87i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (-0.809 - 6.67i)T + (-57.2 + 14.1i)T^{2} \) |
| 61 | \( 1 + (-4.24 - 2.22i)T + (34.6 + 50.2i)T^{2} \) |
| 67 | \( 1 + (-8.50 - 7.53i)T + (8.07 + 66.5i)T^{2} \) |
| 71 | \( 1 + (-3.35 - 4.85i)T + (-25.1 + 66.3i)T^{2} \) |
| 73 | \( 1 + (-1.08 - 8.95i)T + (-70.8 + 17.4i)T^{2} \) |
| 83 | \( 1 + (-2.05 + 16.8i)T + (-80.5 - 19.8i)T^{2} \) |
| 89 | \( 1 + (2.37 - 3.43i)T + (-31.5 - 83.2i)T^{2} \) |
| 97 | \( 1 + (4.91 + 2.58i)T + (55.1 + 79.8i)T^{2} \) |
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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
−13.18160877245071695868226978553, −11.63510071612896983615162234696, −10.70580683514113981618156588915, −10.03175525619649371221543509622, −8.333024184097821390544738691179, −7.71021739784734778574857286176, −6.98032274588469761393578988136, −5.21584293848748543484889635813, −3.94974214960404412228714821304, −1.43970970673597414081081328556,
1.92414727911601190659258032967, 3.66435919193524777197646124703, 5.09177330018918098835878896027, 6.66396152273065541924024464867, 8.132449281720763909794333379132, 8.971496233218808387730034172389, 9.586221960060177420054222670183, 11.23808728500742176616056831455, 11.79693094990276346120554018713, 12.44350183167725724830386662919
