| L(s) = 1 | + (−0.417 + 0.350i)2-s + (−0.766 − 0.642i)3-s + (−0.295 + 1.67i)4-s + (−3.97 + 1.44i)5-s + 0.545·6-s + (−0.377 + 0.137i)7-s + (−1.00 − 1.74i)8-s + (0.173 + 0.984i)9-s + (1.15 − 1.99i)10-s + (1.65 + 2.86i)11-s + (1.30 − 1.09i)12-s + (−0.364 + 2.06i)13-s + (0.109 − 0.189i)14-s + (3.97 + 1.44i)15-s + (−2.16 − 0.788i)16-s + (−0.819 − 4.64i)17-s + ⋯ |
| L(s) = 1 | + (−0.295 + 0.247i)2-s + (−0.442 − 0.371i)3-s + (−0.147 + 0.838i)4-s + (−1.77 + 0.647i)5-s + 0.222·6-s + (−0.142 + 0.0519i)7-s + (−0.356 − 0.617i)8-s + (0.0578 + 0.328i)9-s + (0.364 − 0.632i)10-s + (0.498 + 0.863i)11-s + (0.376 − 0.315i)12-s + (−0.101 + 0.573i)13-s + (0.0292 − 0.0507i)14-s + (1.02 + 0.373i)15-s + (−0.541 − 0.197i)16-s + (−0.198 − 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 - 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.128531 + 0.370858i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.128531 + 0.370858i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.766 + 0.642i)T \) |
| 37 | \( 1 + (-5.01 - 3.44i)T \) |
| good | 2 | \( 1 + (0.417 - 0.350i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (3.97 - 1.44i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.377 - 0.137i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.65 - 2.86i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.364 - 2.06i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (0.819 + 4.64i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (-2.26 - 1.90i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (2.90 - 5.02i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.10 - 5.37i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.38T + 31T^{2} \) |
| 41 | \( 1 + (0.156 - 0.888i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 + (-0.622 + 1.07i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.37 - 1.59i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.643 - 0.234i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.687 - 3.89i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-6.03 + 2.19i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-7.55 - 6.34i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + 0.542T + 73T^{2} \) |
| 79 | \( 1 + (8.97 - 3.26i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.60 - 9.07i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (2.83 + 1.03i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-9.20 + 15.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−14.10951545437398307851832542984, −12.67014761034855653031837230412, −11.79823218459841108866586319676, −11.46295975547786107508321735098, −9.716680314483081269284536830920, −8.328270751898131580405292939005, −7.31139890075294945299090309983, −6.87643916046195591561061347263, −4.52005778077247527947828412835, −3.31672243382641204183207943461,
0.51663725903055538563233343059, 3.75400186744767474167497958968, 4.93764933968340573827796783206, 6.33366519321345831188829743488, 8.073024181942589254481791979090, 8.873242197834941759769345553085, 10.23811250520495619205772851081, 11.22422217125600490940414537632, 11.84772880635872047417184263066, 13.01851844071323446268593225123
