L(s) = 1 | + (1.55 + 1.12i)3-s + (−0.809 + 0.587i)4-s + (0.256 − 0.790i)5-s + (0.828 + 2.55i)9-s − 1.91·12-s + (1.28 − 0.937i)15-s + (0.309 − 0.951i)16-s + (0.256 + 0.790i)20-s − 0.284·23-s + (0.250 + 0.182i)25-s + (−0.997 + 3.07i)27-s + (−0.0879 − 0.270i)31-s + (−2.17 − 1.57i)36-s + (−1.36 + 0.988i)37-s + 2.22·45-s + ⋯ |
L(s) = 1 | + (1.55 + 1.12i)3-s + (−0.809 + 0.587i)4-s + (0.256 − 0.790i)5-s + (0.828 + 2.55i)9-s − 1.91·12-s + (1.28 − 0.937i)15-s + (0.309 − 0.951i)16-s + (0.256 + 0.790i)20-s − 0.284·23-s + (0.250 + 0.182i)25-s + (−0.997 + 3.07i)27-s + (−0.0879 − 0.270i)31-s + (−2.17 − 1.57i)36-s + (−1.36 + 0.988i)37-s + 2.22·45-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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.550852201\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.550852201\) |
\(L(1)\) |
|
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 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (-1.55 - 1.12i)T + (0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.256 + 0.790i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + 0.284T + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.0879 + 0.270i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (1.36 - 0.988i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (1.36 + 0.988i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.404 + 1.24i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-1.05 + 0.769i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - 0.830T + T^{2} \) |
| 71 | \( 1 + (-0.519 + 1.60i)T + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + 1.30T + T^{2} \) |
| 97 | \( 1 + (0.0879 + 0.270i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−9.741830018137061605811661737284, −9.124859189022921177385912624149, −8.379041704970352909613708680935, −8.189573950572190395034200406317, −7.02211904598864961981066119476, −5.19571148218758789466018823200, −4.83565327223038771524849481626, −3.80851654674614514821832609728, −3.26012662545380762124376944957, −1.98029049798854586750929716549,
1.33702279595459545319028097007, 2.40598913735417001545858206503, 3.32355871909510848667761811881, 4.27039101285621516116562469273, 5.70548428846461664327741336017, 6.59866116327131258856699959657, 7.24403903728765565001032047056, 8.167867136041677071722587813970, 8.771688618600680938757988689194, 9.497001988592914605303256902101