| L(s) = 1 | + (−0.270 − 1.70i)2-s + (−3.36 + 0.264i)3-s + (−0.947 + 0.307i)4-s + (−0.735 + 1.20i)5-s + (1.36 + 5.67i)6-s + (−0.137 + 1.74i)7-s + (−0.789 − 1.54i)8-s + (8.26 − 1.30i)9-s + (2.25 + 0.932i)10-s + (2.23 + 2.44i)11-s + (3.10 − 1.28i)12-s + (−1.59 + 2.19i)13-s + (3.02 − 0.238i)14-s + (2.15 − 4.22i)15-s + (−4.04 + 2.93i)16-s + (3.65 + 1.90i)17-s + ⋯ |
| L(s) = 1 | + (−0.191 − 1.20i)2-s + (−1.94 + 0.152i)3-s + (−0.473 + 0.153i)4-s + (−0.328 + 0.536i)5-s + (0.556 + 2.31i)6-s + (−0.0520 + 0.660i)7-s + (−0.278 − 0.547i)8-s + (2.75 − 0.436i)9-s + (0.711 + 0.294i)10-s + (0.674 + 0.738i)11-s + (0.895 − 0.370i)12-s + (−0.441 + 0.607i)13-s + (0.808 − 0.0636i)14-s + (0.556 − 1.09i)15-s + (−1.01 + 0.734i)16-s + (0.886 + 0.463i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.451441 + 0.0829509i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.451441 + 0.0829509i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-2.23 - 2.44i)T \) |
| 17 | \( 1 + (-3.65 - 1.90i)T \) |
| good | 2 | \( 1 + (0.270 + 1.70i)T + (-1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (3.36 - 0.264i)T + (2.96 - 0.469i)T^{2} \) |
| 5 | \( 1 + (0.735 - 1.20i)T + (-2.26 - 4.45i)T^{2} \) |
| 7 | \( 1 + (0.137 - 1.74i)T + (-6.91 - 1.09i)T^{2} \) |
| 13 | \( 1 + (1.59 - 2.19i)T + (-4.01 - 12.3i)T^{2} \) |
| 19 | \( 1 + (2.90 - 1.48i)T + (11.1 - 15.3i)T^{2} \) |
| 23 | \( 1 + (-0.513 + 1.23i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-0.748 - 0.875i)T + (-4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (2.39 - 9.95i)T + (-27.6 - 14.0i)T^{2} \) |
| 37 | \( 1 + (4.10 - 3.51i)T + (5.78 - 36.5i)T^{2} \) |
| 41 | \( 1 + (-4.25 + 4.97i)T + (-6.41 - 40.4i)T^{2} \) |
| 43 | \( 1 + (-0.377 + 0.377i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.05 + 2.29i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.334 + 2.11i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (4.65 + 2.36i)T + (34.6 + 47.7i)T^{2} \) |
| 61 | \( 1 + (-2.10 + 0.505i)T + (54.3 - 27.6i)T^{2} \) |
| 67 | \( 1 - 6.67T + 67T^{2} \) |
| 71 | \( 1 + (5.15 + 3.16i)T + (32.2 + 63.2i)T^{2} \) |
| 73 | \( 1 + (1.71 + 2.00i)T + (-11.4 + 72.1i)T^{2} \) |
| 79 | \( 1 + (-1.10 + 0.678i)T + (35.8 - 70.3i)T^{2} \) |
| 83 | \( 1 + (-13.2 - 2.10i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + 1.27iT - 89T^{2} \) |
| 97 | \( 1 + (4.65 + 1.11i)T + (86.4 + 44.0i)T^{2} \) |
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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
−12.22176217471197671457370316940, −11.72327750525119417642876196531, −10.77173794861238475733086990559, −10.23338056302808383608652676556, −9.225634105817374041648987083871, −7.06126855770742196133787234639, −6.37601101941184238770318704073, −5.02133284579018588937985446961, −3.73660509358015695691078560036, −1.62845207863929954459884011966,
0.58098099693231686196178347600, 4.36030336393189386578097613342, 5.43499562384746087939281112222, 6.21002883307536078886722851863, 7.15401442372988180181321650662, 7.987271166728781494476857023458, 9.553512354502928296111153361822, 10.78255312285717862178396324229, 11.56988013445407486193923636753, 12.35557322958641825322665971997
