| L(s) = 1 | + (0.836 + 0.503i)2-s + (0.370 − 0.928i)3-s + (−0.490 − 0.924i)4-s + (1.49 + 0.162i)5-s + (0.777 − 0.590i)6-s + (−0.307 − 0.103i)7-s + (0.161 − 2.97i)8-s + (−0.725 − 0.687i)9-s + (1.16 + 0.886i)10-s + (1.46 + 2.16i)11-s + (−1.04 + 0.113i)12-s + (−0.109 + 0.103i)13-s + (−0.205 − 0.241i)14-s + (0.702 − 1.32i)15-s + (0.455 − 0.672i)16-s + (0.424 − 0.142i)17-s + ⋯ |
| L(s) = 1 | + (0.591 + 0.355i)2-s + (0.213 − 0.536i)3-s + (−0.245 − 0.462i)4-s + (0.667 + 0.0725i)5-s + (0.317 − 0.241i)6-s + (−0.116 − 0.0391i)7-s + (0.0569 − 1.05i)8-s + (−0.241 − 0.229i)9-s + (0.368 + 0.280i)10-s + (0.442 + 0.652i)11-s + (−0.300 + 0.0326i)12-s + (−0.0304 + 0.0288i)13-s + (−0.0547 − 0.0645i)14-s + (0.181 − 0.342i)15-s + (0.113 − 0.168i)16-s + (0.102 − 0.0346i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59890 - 0.349538i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59890 - 0.349538i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.370 + 0.928i)T \) |
| 59 | \( 1 + (-7.13 - 2.83i)T \) |
| good | 2 | \( 1 + (-0.836 - 0.503i)T + (0.936 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-1.49 - 0.162i)T + (4.88 + 1.07i)T^{2} \) |
| 7 | \( 1 + (0.307 + 0.103i)T + (5.57 + 4.23i)T^{2} \) |
| 11 | \( 1 + (-1.46 - 2.16i)T + (-4.07 + 10.2i)T^{2} \) |
| 13 | \( 1 + (0.109 - 0.103i)T + (0.703 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.424 + 0.142i)T + (13.5 - 10.2i)T^{2} \) |
| 19 | \( 1 + (1.13 - 6.93i)T + (-18.0 - 6.06i)T^{2} \) |
| 23 | \( 1 + (-0.663 + 2.38i)T + (-19.7 - 11.8i)T^{2} \) |
| 29 | \( 1 + (2.37 - 1.43i)T + (13.5 - 25.6i)T^{2} \) |
| 31 | \( 1 + (-1.24 - 7.58i)T + (-29.3 + 9.89i)T^{2} \) |
| 37 | \( 1 + (-0.282 - 5.20i)T + (-36.7 + 4.00i)T^{2} \) |
| 41 | \( 1 + (2.85 + 10.2i)T + (-35.1 + 21.1i)T^{2} \) |
| 43 | \( 1 + (-1.36 + 2.01i)T + (-15.9 - 39.9i)T^{2} \) |
| 47 | \( 1 + (1.67 - 0.182i)T + (45.9 - 10.1i)T^{2} \) |
| 53 | \( 1 + (-6.13 + 4.66i)T + (14.1 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.59 + 2.16i)T + (28.5 + 53.8i)T^{2} \) |
| 67 | \( 1 + (-0.0149 + 0.276i)T + (-66.6 - 7.24i)T^{2} \) |
| 71 | \( 1 + (-3.20 + 0.348i)T + (69.3 - 15.2i)T^{2} \) |
| 73 | \( 1 + (-3.49 - 4.11i)T + (-11.8 + 72.0i)T^{2} \) |
| 79 | \( 1 + (-1.24 - 3.11i)T + (-57.3 + 54.3i)T^{2} \) |
| 83 | \( 1 + (2.08 + 0.964i)T + (53.7 + 63.2i)T^{2} \) |
| 89 | \( 1 + (-10.2 + 6.15i)T + (41.6 - 78.6i)T^{2} \) |
| 97 | \( 1 + (0.0367 - 0.0433i)T + (-15.6 - 95.7i)T^{2} \) |
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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
−12.79316219995426983706509606099, −12.05593724286160232921284328845, −10.39322839907230035830800993244, −9.735202101888869746437620561998, −8.550105759655577935969179553760, −7.08595036537093282089323676704, −6.24783291195369900181289354180, −5.23153048241043508988440787927, −3.77731621112306068638558653010, −1.74012901122889861290638064746,
2.53696938341540607816462757100, 3.79203552357257208611547935305, 4.95687308281844822585597975183, 6.11571695661464941812285291870, 7.76625676025778803570957143288, 8.941432273791733004264833193345, 9.625281711365004227919999704030, 11.05628629833314010082771500893, 11.68885245749536512991407223937, 13.10683758545812608560428700483
