| 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
−12.44350183167725724830386662919, −11.79693094990276346120554018713, −11.23808728500742176616056831455, −9.586221960060177420054222670183, −8.971496233218808387730034172389, −8.132449281720763909794333379132, −6.66396152273065541924024464867, −5.09177330018918098835878896027, −3.66435919193524777197646124703, −1.92414727911601190659258032967,
1.43970970673597414081081328556, 3.94974214960404412228714821304, 5.21584293848748543484889635813, 6.98032274588469761393578988136, 7.71021739784734778574857286176, 8.333024184097821390544738691179, 10.03175525619649371221543509622, 10.70580683514113981618156588915, 11.63510071612896983615162234696, 13.18160877245071695868226978553
