L(s) = 1 | + (0.519 − 1.60i)3-s + (0.309 + 0.951i)4-s + (1.05 − 0.769i)5-s + (−1.48 − 1.07i)9-s + 1.68·12-s + (−0.680 − 2.09i)15-s + (−0.809 + 0.587i)16-s + (1.05 + 0.769i)20-s − 1.91·23-s + (0.221 − 0.680i)25-s + (−1.13 + 0.821i)27-s + (1.55 + 1.12i)31-s + (0.565 − 1.74i)36-s + (0.256 + 0.790i)37-s − 2.39·45-s + ⋯ |
L(s) = 1 | + (0.519 − 1.60i)3-s + (0.309 + 0.951i)4-s + (1.05 − 0.769i)5-s + (−1.48 − 1.07i)9-s + 1.68·12-s + (−0.680 − 2.09i)15-s + (−0.809 + 0.587i)16-s + (1.05 + 0.769i)20-s − 1.91·23-s + (0.221 − 0.680i)25-s + (−1.13 + 0.821i)27-s + (1.55 + 1.12i)31-s + (0.565 − 1.74i)36-s + (0.256 + 0.790i)37-s − 2.39·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.535989965\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.535989965\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 3 | \( 1 + (-0.519 + 1.60i)T + (-0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (-1.05 + 0.769i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + 1.91T + T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-1.55 - 1.12i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.256 - 0.790i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.256 + 0.790i)T + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.230 - 0.167i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.0879 + 0.270i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + 1.30T + T^{2} \) |
| 71 | \( 1 + (0.672 - 0.488i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + 0.284T + T^{2} \) |
| 97 | \( 1 + (-1.55 - 1.12i)T + (0.309 + 0.951i)T^{2} \) |
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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
−9.404610174796893004876199143311, −8.457400656318352547034082273431, −8.187776847901538094899820123022, −7.27412056369696162497196935739, −6.46422490233456165069011711778, −5.86415004892171914931522701250, −4.52171489747595964815957243454, −3.14633781865876696699388297217, −2.23662004123553542176174588292, −1.44555140690865457603447390606,
2.06059751667065391828083876081, 2.81617293040663643274586352083, 4.03846525659286494854444504653, 4.88360479320888780899155856469, 5.94369303278102486620494349431, 6.22849376082595726787915407763, 7.61778812025058756689646618403, 8.710171748698903722591594936807, 9.607664686126343277911923132058, 9.965618505806687349636692354041