| L(s) = 1 | + (0.888 − 0.458i)2-s + (−0.786 + 0.618i)3-s + (0.580 − 0.814i)4-s + (0.258 − 0.246i)5-s + (−0.415 + 0.909i)6-s + (2.35 − 1.21i)7-s + (0.142 − 0.989i)8-s + (0.235 − 0.971i)9-s + (0.116 − 0.337i)10-s + (−5.84 − 3.01i)11-s + (0.0475 + 0.998i)12-s + (−3.08 − 3.55i)13-s + (1.53 − 2.15i)14-s + (−0.0508 + 0.353i)15-s + (−0.327 − 0.945i)16-s + (−4.16 + 0.397i)17-s + ⋯ |
| L(s) = 1 | + (0.628 − 0.324i)2-s + (−0.453 + 0.356i)3-s + (0.290 − 0.407i)4-s + (0.115 − 0.110i)5-s + (−0.169 + 0.371i)6-s + (0.889 − 0.457i)7-s + (0.0503 − 0.349i)8-s + (0.0785 − 0.323i)9-s + (0.0369 − 0.106i)10-s + (−1.76 − 0.908i)11-s + (0.0137 + 0.288i)12-s + (−0.855 − 0.986i)13-s + (0.410 − 0.575i)14-s + (−0.0131 + 0.0913i)15-s + (−0.0817 − 0.236i)16-s + (−1.01 + 0.0965i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.876570 - 1.30967i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.876570 - 1.30967i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.888 + 0.458i)T \) |
| 3 | \( 1 + (0.786 - 0.618i)T \) |
| 7 | \( 1 + (-2.35 + 1.21i)T \) |
| 23 | \( 1 + (-0.315 + 4.78i)T \) |
| good | 5 | \( 1 + (-0.258 + 0.246i)T + (0.237 - 4.99i)T^{2} \) |
| 11 | \( 1 + (5.84 + 3.01i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (3.08 + 3.55i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (4.16 - 0.397i)T + (16.6 - 3.21i)T^{2} \) |
| 19 | \( 1 + (-7.05 - 0.673i)T + (18.6 + 3.59i)T^{2} \) |
| 29 | \( 1 + (-1.12 + 2.46i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (4.77 - 1.91i)T + (22.4 - 21.3i)T^{2} \) |
| 37 | \( 1 + (-2.71 + 11.1i)T + (-32.8 - 16.9i)T^{2} \) |
| 41 | \( 1 + (-2.74 - 0.806i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.545 - 3.79i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (0.221 - 0.383i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.713 - 0.137i)T + (49.2 + 19.6i)T^{2} \) |
| 59 | \( 1 + (-0.725 + 2.09i)T + (-46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (-3.24 - 2.55i)T + (14.3 + 59.2i)T^{2} \) |
| 67 | \( 1 + (-0.0152 + 0.320i)T + (-66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (11.5 + 7.39i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-7.15 + 10.0i)T + (-23.8 - 68.9i)T^{2} \) |
| 79 | \( 1 + (15.4 - 2.98i)T + (73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (-8.65 + 2.54i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-11.7 - 4.72i)T + (64.4 + 61.4i)T^{2} \) |
| 97 | \( 1 + (-4.48 - 1.31i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.11889127308991597961872727136, −9.054201289072566229472650630670, −7.85602504207599818721735090823, −7.36565503309184822675056286074, −5.86569433813610056847513745306, −5.25958976819992676791216710552, −4.68044594711505690760266685663, −3.38639409332951867861043733456, −2.36604696423044150057642532427, −0.59725320749647176625924543519,
1.92652573026605111643953513598, 2.74483022964617559452111584138, 4.53613121108305458387874843164, 5.03087389288544973378974618454, 5.76765394919652990190973211789, 7.07786922153915743789081993430, 7.43458963874991238679301906849, 8.345046077982935872866490141182, 9.512596137755589544071751233573, 10.36570843997256893255609714197
