| L(s) = 1 | + (−0.122 + 1.63i)2-s + (0.324 + 1.70i)3-s + (−0.690 − 0.104i)4-s + (0.354 − 1.55i)5-s + (−2.82 + 0.322i)6-s + (2.24 + 1.40i)7-s + (−0.475 + 2.08i)8-s + (−2.78 + 1.10i)9-s + (2.49 + 0.771i)10-s + (1.86 − 0.899i)11-s + (−0.0469 − 1.20i)12-s + (−0.129 + 1.72i)13-s + (−2.57 + 3.49i)14-s + (2.75 + 0.0991i)15-s + (−4.69 − 1.44i)16-s + (−3.41 + 0.514i)17-s + ⋯ |
| L(s) = 1 | + (−0.0867 + 1.15i)2-s + (0.187 + 0.982i)3-s + (−0.345 − 0.0520i)4-s + (0.158 − 0.694i)5-s + (−1.15 + 0.131i)6-s + (0.847 + 0.531i)7-s + (−0.168 + 0.737i)8-s + (−0.929 + 0.368i)9-s + (0.790 + 0.243i)10-s + (0.563 − 0.271i)11-s + (−0.0135 − 0.348i)12-s + (−0.0358 + 0.477i)13-s + (−0.689 + 0.934i)14-s + (0.711 + 0.0256i)15-s + (−1.17 − 0.361i)16-s + (−0.828 + 0.124i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.322608 + 1.55560i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.322608 + 1.55560i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.324 - 1.70i)T \) |
| 7 | \( 1 + (-2.24 - 1.40i)T \) |
| good | 2 | \( 1 + (0.122 - 1.63i)T + (-1.97 - 0.298i)T^{2} \) |
| 5 | \( 1 + (-0.354 + 1.55i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (-1.86 + 0.899i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (0.129 - 1.72i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (3.41 - 0.514i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-1.13 - 1.97i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.552 - 0.692i)T + (-5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (-0.740 + 1.88i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (-0.776 - 1.34i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.883 + 2.25i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (7.62 + 7.07i)T + (3.06 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-6.98 + 6.48i)T + (3.21 - 42.8i)T^{2} \) |
| 47 | \( 1 + (-0.861 + 11.4i)T + (-46.4 - 7.00i)T^{2} \) |
| 53 | \( 1 + (-1.74 - 4.45i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-1.23 + 1.14i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (2.31 - 0.349i)T + (58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (-5.83 - 10.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.05 - 1.32i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-13.1 + 8.96i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (0.697 - 1.20i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.451 - 6.02i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (-0.761 - 10.1i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (1.75 + 3.04i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.48205280667523819267471798746, −10.58047191504065048847975236562, −9.192213428296058893168614724664, −8.787343509118588326239807325884, −8.079973733400806415914623244413, −6.83865324526773290452269050621, −5.65494086153603630010228418893, −5.08122128214514424650860621368, −4.01277312335080725969625087447, −2.19170938628128214333296700378,
1.09318953997397391838665384775, 2.26301542490498054853726127710, 3.21902590206335840380586696894, 4.61346580851294453020305579605, 6.33458499624810873615605525196, 6.98192540849638148035808154099, 7.956443277355375300362340837347, 9.064784274268278763393549254432, 10.08399726998502802960441535293, 11.10370316757174222225755316561
