| L(s) = 1 | + (−0.280 + 0.649i)2-s + (1.02 + 1.09i)4-s + (0.199 + 3.42i)5-s + (2.63 + 0.623i)7-s + (−2.32 + 0.846i)8-s + (−2.28 − 0.831i)10-s + (1.23 + 0.811i)11-s + (5.51 + 0.644i)13-s + (−1.14 + 1.53i)14-s + (−0.0725 + 1.24i)16-s + (3.86 − 3.24i)17-s + (−3.94 − 3.30i)19-s + (−3.53 + 3.74i)20-s + (−0.873 + 0.574i)22-s + (−1.68 + 0.399i)23-s + ⋯ |
| L(s) = 1 | + (−0.198 + 0.459i)2-s + (0.514 + 0.545i)4-s + (0.0893 + 1.53i)5-s + (0.994 + 0.235i)7-s + (−0.822 + 0.299i)8-s + (−0.722 − 0.262i)10-s + (0.372 + 0.244i)11-s + (1.52 + 0.178i)13-s + (−0.305 + 0.410i)14-s + (−0.0181 + 0.311i)16-s + (0.937 − 0.786i)17-s + (−0.904 − 0.758i)19-s + (−0.790 + 0.837i)20-s + (−0.186 + 0.122i)22-s + (−0.351 + 0.0833i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.494 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.917748 + 1.57870i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.917748 + 1.57870i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (0.280 - 0.649i)T + (-1.37 - 1.45i)T^{2} \) |
| 5 | \( 1 + (-0.199 - 3.42i)T + (-4.96 + 0.580i)T^{2} \) |
| 7 | \( 1 + (-2.63 - 0.623i)T + (6.25 + 3.14i)T^{2} \) |
| 11 | \( 1 + (-1.23 - 0.811i)T + (4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-5.51 - 0.644i)T + (12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (-3.86 + 3.24i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (3.94 + 3.30i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (1.68 - 0.399i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (1.28 + 1.72i)T + (-8.31 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-1.09 + 3.64i)T + (-25.9 - 17.0i)T^{2} \) |
| 37 | \( 1 + (0.891 + 5.05i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-1.78 - 4.14i)T + (-28.1 + 29.8i)T^{2} \) |
| 43 | \( 1 + (-0.736 + 0.370i)T + (25.6 - 34.4i)T^{2} \) |
| 47 | \( 1 + (1.92 + 6.43i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (2.58 + 4.48i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.95 - 1.94i)T + (23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (-3.71 + 3.94i)T + (-3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (3.91 - 5.25i)T + (-19.2 - 64.1i)T^{2} \) |
| 71 | \( 1 + (3.30 + 1.20i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.668 + 0.243i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (6.21 - 14.4i)T + (-54.2 - 57.4i)T^{2} \) |
| 83 | \( 1 + (2.35 - 5.45i)T + (-56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (-3.78 + 1.37i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.700 + 12.0i)T + (-96.3 - 11.2i)T^{2} \) |
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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
−11.04025043563037939388042662588, −9.841250251508258892631203695296, −8.709656494314590472450528172721, −7.974916574874192467061067296018, −7.15200132233492851742565522719, −6.47343213870013363362935586880, −5.67204937391011934893812968595, −4.06160467454896559685430834988, −3.04740309597235329158119095689, −2.00095508493784119149539422363,
1.19036796865031402151561849732, 1.61966671849575575483954966663, 3.56119341469460255519153152210, 4.61699229021436067801712799906, 5.69244260840502782216564055471, 6.26187225998042868859632192495, 7.86459768236929462498539720805, 8.511072177378501917857649769495, 9.158204885587538173960659826286, 10.31594258931775933162734621081
