| L(s) = 1 | + (1.08 + 0.910i)2-s + (−0.131 + 1.72i)3-s + (0.343 + 1.97i)4-s + (2.24 + 0.672i)5-s + (−1.71 + 1.74i)6-s + (−1.74 − 0.754i)7-s + (−1.42 + 2.44i)8-s + (−2.96 − 0.453i)9-s + (1.81 + 2.77i)10-s + (0.310 + 0.0736i)11-s + (−3.44 + 0.334i)12-s + (5.18 − 3.40i)13-s + (−1.20 − 2.40i)14-s + (−1.45 + 3.78i)15-s + (−3.76 + 1.35i)16-s + (−0.544 + 1.49i)17-s + ⋯ |
| L(s) = 1 | + (0.765 + 0.643i)2-s + (−0.0757 + 0.997i)3-s + (0.171 + 0.985i)4-s + (1.00 + 0.300i)5-s + (−0.699 + 0.714i)6-s + (−0.661 − 0.285i)7-s + (−0.502 + 0.864i)8-s + (−0.988 − 0.151i)9-s + (0.575 + 0.876i)10-s + (0.0937 + 0.0222i)11-s + (−0.995 + 0.0965i)12-s + (1.43 − 0.945i)13-s + (−0.322 − 0.643i)14-s + (−0.375 + 0.978i)15-s + (−0.941 + 0.338i)16-s + (−0.132 + 0.362i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00189 + 1.78491i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00189 + 1.78491i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.08 - 0.910i)T \) |
| 3 | \( 1 + (0.131 - 1.72i)T \) |
| good | 5 | \( 1 + (-2.24 - 0.672i)T + (4.17 + 2.74i)T^{2} \) |
| 7 | \( 1 + (1.74 + 0.754i)T + (4.80 + 5.09i)T^{2} \) |
| 11 | \( 1 + (-0.310 - 0.0736i)T + (9.82 + 4.93i)T^{2} \) |
| 13 | \( 1 + (-5.18 + 3.40i)T + (5.14 - 11.9i)T^{2} \) |
| 17 | \( 1 + (0.544 - 1.49i)T + (-13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (0.703 + 1.93i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (1.82 + 4.24i)T + (-15.7 + 16.7i)T^{2} \) |
| 29 | \( 1 + (-7.16 - 0.417i)T + (28.8 + 3.36i)T^{2} \) |
| 31 | \( 1 + (0.769 - 6.58i)T + (-30.1 - 7.14i)T^{2} \) |
| 37 | \( 1 + (6.30 - 5.29i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (0.0778 - 0.154i)T + (-24.4 - 32.8i)T^{2} \) |
| 43 | \( 1 + (-4.04 - 3.81i)T + (2.50 + 42.9i)T^{2} \) |
| 47 | \( 1 + (-12.3 + 1.44i)T + (45.7 - 10.8i)T^{2} \) |
| 53 | \( 1 + (-0.326 + 0.188i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.0 + 2.37i)T + (52.7 - 26.4i)T^{2} \) |
| 61 | \( 1 + (7.69 + 10.3i)T + (-17.4 + 58.4i)T^{2} \) |
| 67 | \( 1 + (5.70 - 0.332i)T + (66.5 - 7.77i)T^{2} \) |
| 71 | \( 1 + (1.46 + 8.28i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (0.0180 - 0.102i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.05 + 2.10i)T + (-47.1 + 63.3i)T^{2} \) |
| 83 | \( 1 + (0.0362 - 0.0182i)T + (49.5 - 66.5i)T^{2} \) |
| 89 | \( 1 + (1.09 + 0.193i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (0.749 + 2.50i)T + (-81.0 + 53.3i)T^{2} \) |
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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
−12.06913839141729892162561788011, −10.71759948821326595765406978568, −10.31539492692027925357670684186, −9.016622027905975334745337577748, −8.269728155638165638568583037133, −6.56333808860058413626009993838, −6.09033634116479030646359283281, −5.03334906652633905707848057181, −3.78989434212334034654432558508, −2.83349280198408838737563738127,
1.38818090621136007838554257258, 2.48831866016128367573174090738, 3.94710701262045190573400662478, 5.70226834601060452731160953380, 6.02894712743542785970522303223, 7.06582480747225609869871411353, 8.758754735240202302350246367336, 9.449283720856446277078901415651, 10.58683797503282928138968360697, 11.59798488201152931611559942183
