| L(s) = 1 | + (−0.173 + 2.31i)2-s + (−0.732 + 1.56i)3-s + (−3.32 − 0.501i)4-s + (−0.445 + 1.95i)5-s + (−3.49 − 1.96i)6-s + (0.693 − 2.55i)7-s + (0.704 − 3.08i)8-s + (−1.92 − 2.29i)9-s + (−4.42 − 1.36i)10-s + (−2.98 + 1.43i)11-s + (3.22 − 4.85i)12-s + (0.277 − 3.69i)13-s + (5.77 + 2.04i)14-s + (−2.73 − 2.12i)15-s + (0.572 + 0.176i)16-s + (−5.17 + 0.779i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 1.63i)2-s + (−0.422 + 0.906i)3-s + (−1.66 − 0.250i)4-s + (−0.199 + 0.872i)5-s + (−1.42 − 0.801i)6-s + (0.262 − 0.965i)7-s + (0.248 − 1.09i)8-s + (−0.642 − 0.766i)9-s + (−1.40 − 0.431i)10-s + (−0.901 + 0.434i)11-s + (0.931 − 1.40i)12-s + (0.0768 − 1.02i)13-s + (1.54 + 0.546i)14-s + (−0.706 − 0.549i)15-s + (0.143 + 0.0441i)16-s + (−1.25 + 0.189i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.322579 - 0.191022i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.322579 - 0.191022i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.732 - 1.56i)T \) |
| 7 | \( 1 + (-0.693 + 2.55i)T \) |
| good | 2 | \( 1 + (0.173 - 2.31i)T + (-1.97 - 0.298i)T^{2} \) |
| 5 | \( 1 + (0.445 - 1.95i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (2.98 - 1.43i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (-0.277 + 3.69i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (5.17 - 0.779i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-0.930 - 1.61i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.48 - 1.86i)T + (-5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (1.83 - 4.68i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (-0.858 - 1.48i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.33 - 8.50i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (0.0655 + 0.0607i)T + (3.06 + 40.8i)T^{2} \) |
| 43 | \( 1 + (3.16 - 2.93i)T + (3.21 - 42.8i)T^{2} \) |
| 47 | \( 1 + (-0.723 + 9.65i)T + (-46.4 - 7.00i)T^{2} \) |
| 53 | \( 1 + (-0.956 - 2.43i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (9.73 - 9.03i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (6.63 - 0.999i)T + (58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (-3.55 - 6.15i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.33 + 7.94i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (6.80 - 4.64i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-5.12 + 8.87i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.118 + 1.58i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (1.11 + 14.9i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (0.0150 + 0.0261i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.57741773332685587610177798664, −10.53898150392592823531437657073, −10.25362287310533291743912702992, −8.938493194777001251079114537568, −7.972944829031212350570138076084, −7.18548650995418564150130391667, −6.39065450789999909849576988034, −5.30518049887348035781812870085, −4.54323233560797097583481296691, −3.26818691778818847149337961443,
0.24970434819922884343897341012, 1.84310229227244964540963349847, 2.63698965296286405582510363808, 4.38578676591829868526014344042, 5.26290803714060558873136485211, 6.51664784270130398278916479934, 8.012878195811705969549322560711, 8.789583473537085817127221119906, 9.427842552190758534155290710421, 10.94541152944507961923595744778
