L(s) = 1 | + (1.5 − 0.866i)3-s + (−8.29 − 4.79i)5-s + (−0.5 + 6.98i)7-s + (1.5 − 2.59i)9-s + (2.29 + 3.97i)11-s + 7.84i·13-s − 16.5·15-s + (−16.5 + 9.58i)17-s + (14.2 + 8.20i)19-s + (5.29 + 10.9i)21-s + (−12 + 20.7i)23-s + (33.3 + 57.8i)25-s − 5.19i·27-s − 10.5·29-s + (−48.2 + 27.8i)31-s + ⋯ |
L(s) = 1 | + (0.5 − 0.288i)3-s + (−1.65 − 0.958i)5-s + (−0.0714 + 0.997i)7-s + (0.166 − 0.288i)9-s + (0.208 + 0.361i)11-s + 0.603i·13-s − 1.10·15-s + (−0.976 + 0.563i)17-s + (0.747 + 0.431i)19-s + (0.252 + 0.519i)21-s + (−0.521 + 0.903i)23-s + (1.33 + 2.31i)25-s − 0.192i·27-s − 0.365·29-s + (−1.55 + 0.899i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.386525 + 0.538320i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.386525 + 0.538320i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 6.98i)T \) |
good | 5 | \( 1 + (8.29 + 4.79i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-2.29 - 3.97i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 7.84iT - 169T^{2} \) |
| 17 | \( 1 + (16.5 - 9.58i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-14.2 - 8.20i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (12 - 20.7i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 10.5T + 841T^{2} \) |
| 31 | \( 1 + (48.2 - 27.8i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-12.2 + 21.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 48.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 18.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-10.4 - 6.00i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-18.7 - 32.3i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (54.7 - 31.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (96.3 + 55.6i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-31.9 - 55.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 21.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (53.5 - 30.9i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (1.90 - 3.30i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 100. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-85.7 - 49.5i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 63.5iT - 9.40e3T^{2} \) |
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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
−11.90427738701699699623028358508, −10.97197294089830371077135126983, −9.186474542874880835729303714726, −8.933161258957606423587995718381, −7.86812775997552542294110611541, −7.16365741081081569635676104490, −5.62876049485506154671223033074, −4.38300279889728084697984067712, −3.49195086411855691442841683102, −1.72503431728619745228009509142,
0.28973834727422522716543048228, 2.88126746569171385918090468967, 3.76306467633733172666879584856, 4.59378971911858087046125180163, 6.51034439032143094208341305086, 7.46393269432068006642805107332, 7.936785024557267722615977680450, 9.140738913207466427635201758789, 10.36837615155372663532302815059, 11.05313572783309405677200187132