L(s) = 1 | + 1.41i·2-s − 1.00·4-s + 1.41i·11-s − 0.999·16-s − 2.00·22-s − 1.41i·23-s + 25-s − 1.41i·29-s − 1.41i·32-s − 1.41i·44-s + 2.00·46-s + 1.41i·50-s − 1.41i·53-s + 2.00·58-s + 1.00·64-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 1.00·4-s + 1.41i·11-s − 0.999·16-s − 2.00·22-s − 1.41i·23-s + 25-s − 1.41i·29-s − 1.41i·32-s − 1.41i·44-s + 2.00·46-s + 1.41i·50-s − 1.41i·53-s + 2.00·58-s + 1.00·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8415097293\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8415097293\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.41iT - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 2T + T^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−11.75768382820734119807401977288, −10.58591257591691448054745128864, −9.637006426297288684903192565796, −8.669173095535633182753122905473, −7.82066734516852205356545624700, −6.96600112572225974598710138211, −6.28449274811526203539564201486, −5.07896495965005685939838322715, −4.31225387507632891502908836307, −2.36438985690773399053883083814,
1.32837335488096312434276989129, 2.91671596863141148818752592879, 3.64577065757865264279999126722, 5.01233410038919233840908818842, 6.21468501338760857365001128365, 7.44273120123202670099339716443, 8.737212900068718365040170104805, 9.329485489699679464914913646392, 10.51651999613889885513362405791, 10.96857346074114353539039441239