L(s) = 1 | + (0.5 − 0.866i)3-s + (−1.70 − 2.95i)5-s + (−0.499 − 0.866i)9-s + (−0.414 + 0.717i)11-s − 4.24·13-s − 3.41·15-s + (−3.70 + 6.42i)17-s + (−3.41 − 5.91i)19-s + (2.41 + 4.18i)23-s + (−3.32 + 5.76i)25-s − 0.999·27-s + 2.82·29-s + (1.41 − 2.44i)31-s + (0.414 + 0.717i)33-s + (0.828 + 1.43i)37-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.763 − 1.32i)5-s + (−0.166 − 0.288i)9-s + (−0.124 + 0.216i)11-s − 1.17·13-s − 0.881·15-s + (−0.899 + 1.55i)17-s + (−0.783 − 1.35i)19-s + (0.503 + 0.871i)23-s + (−0.665 + 1.15i)25-s − 0.192·27-s + 0.525·29-s + (0.254 − 0.439i)31-s + (0.0721 + 0.124i)33-s + (0.136 + 0.235i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2425530881\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2425530881\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.70 + 2.95i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.414 - 0.717i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + (3.70 - 6.42i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.41 + 5.91i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.41 - 4.18i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + (-1.41 + 2.44i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.828 - 1.43i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + (-2.24 - 3.88i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.24 - 7.34i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.53 + 9.58i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.65 - 9.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + (3.87 - 6.71i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.82 + 11.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (2.87 + 4.98i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 0.242T + 97T^{2} \) |
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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
−8.963097834016151816545897020221, −8.544849305818352039629889262456, −7.70111846801371911632411133226, −6.97390761611863246224838246919, −5.88433798386251881974248718681, −4.67482098570808980575486213072, −4.29078179741364707521276867065, −2.79907694696354914279291197333, −1.56655840831453853250316687495, −0.096669579733851806959884073859,
2.44955167162224465407072878229, 3.05284246044623820653025464842, 4.17022548377157239787902181174, 4.94630944632226708174643445030, 6.28530035786430797223743462655, 7.07396833367428870223029394783, 7.71917801825459096857699989134, 8.624801526350862192453876562586, 9.560988391829320781087929637952, 10.40897950143917754272701755519