L(s) = 1 | + (0.867 + 1.11i)2-s + (−0.494 + 1.93i)4-s + (−0.895 + 1.55i)5-s + (−2.08 + 1.20i)7-s + (−2.59 + 1.12i)8-s + (−2.50 + 0.345i)10-s + (1.36 − 0.790i)11-s + (5.35 + 3.09i)13-s + (−3.15 − 1.28i)14-s + (−3.51 − 1.91i)16-s − 3.69i·17-s + 3.12·19-s + (−2.56 − 2.50i)20-s + (2.07 + 0.843i)22-s + (1.36 − 2.35i)23-s + ⋯ |
L(s) = 1 | + (0.613 + 0.789i)2-s + (−0.247 + 0.968i)4-s + (−0.400 + 0.693i)5-s + (−0.789 + 0.455i)7-s + (−0.916 + 0.398i)8-s + (−0.793 + 0.109i)10-s + (0.412 − 0.238i)11-s + (1.48 + 0.857i)13-s + (−0.843 − 0.343i)14-s + (−0.877 − 0.479i)16-s − 0.897i·17-s + 0.717·19-s + (−0.572 − 0.559i)20-s + (0.441 + 0.179i)22-s + (0.283 − 0.491i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.689907 + 1.22467i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.689907 + 1.22467i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.867 - 1.11i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.895 - 1.55i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (2.08 - 1.20i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.36 + 0.790i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.35 - 3.09i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3.69iT - 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 + (-1.36 + 2.35i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.55 - 4.42i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.95 + 3.43i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.24iT - 37T^{2} \) |
| 41 | \( 1 + (-5.32 - 3.07i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.452 + 0.783i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.88 + 8.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 7.05T + 53T^{2} \) |
| 59 | \( 1 + (-6.10 - 3.52i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.05 - 1.76i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.03 + 1.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.31T + 71T^{2} \) |
| 73 | \( 1 - 0.631T + 73T^{2} \) |
| 79 | \( 1 + (-7.82 + 4.51i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (13.5 - 7.82i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 1.16iT - 89T^{2} \) |
| 97 | \( 1 + (6.72 + 11.6i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.82132307482297650032448840744, −11.73512701661155897259345418429, −11.05724299216048675203914631520, −9.374121689461764676377875019547, −8.662329299693646355275058911199, −7.24882539024761439891928949617, −6.57461237892193843644102786124, −5.55290102403316960722643452950, −3.95584200430514196484730046926, −3.02683312250912332990950407405,
1.12096017564893920421714367734, 3.30459226541255685651313188307, 4.15716507546979393411359897667, 5.54982941692291816762402581676, 6.58106865057804046758441638364, 8.196036618814802314580793107936, 9.241124746027704811051257057490, 10.24549280390966512755167168053, 11.10650641952619791792098550535, 12.13916050048435805778210619898