L(s) = 1 | + (−2.23 − 3.07i)5-s + (0.349 − 0.113i)7-s + (2.97 − 1.46i)11-s + (−0.557 + 0.767i)13-s + (2.77 − 2.01i)17-s + (−4.05 − 1.31i)19-s − 4.96i·23-s + (−2.90 + 8.94i)25-s + (−0.767 − 2.36i)29-s + (2.84 + 2.06i)31-s + (−1.12 − 0.820i)35-s + (−2.21 − 6.83i)37-s + (−0.840 + 2.58i)41-s + 1.88i·43-s + (−0.0195 − 0.00636i)47-s + ⋯ |
L(s) = 1 | + (−0.997 − 1.37i)5-s + (0.132 − 0.0429i)7-s + (0.897 − 0.440i)11-s + (−0.154 + 0.212i)13-s + (0.673 − 0.489i)17-s + (−0.929 − 0.302i)19-s − 1.03i·23-s + (−0.581 + 1.78i)25-s + (−0.142 − 0.438i)29-s + (0.510 + 0.370i)31-s + (−0.190 − 0.138i)35-s + (−0.364 − 1.12i)37-s + (−0.131 + 0.403i)41-s + 0.287i·43-s + (−0.00285 − 0.000928i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8748748393\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8748748393\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-2.97 + 1.46i)T \) |
good | 5 | \( 1 + (2.23 + 3.07i)T + (-1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.349 + 0.113i)T + (5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (0.557 - 0.767i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.77 + 2.01i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (4.05 + 1.31i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 4.96iT - 23T^{2} \) |
| 29 | \( 1 + (0.767 + 2.36i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.84 - 2.06i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.21 + 6.83i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.840 - 2.58i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 1.88iT - 43T^{2} \) |
| 47 | \( 1 + (0.0195 + 0.00636i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (3.25 - 4.48i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (6.29 - 2.04i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (5.73 + 7.88i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 4.46T + 67T^{2} \) |
| 71 | \( 1 + (6.06 + 8.35i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.18 - 1.35i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (6.42 - 8.84i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (7.42 - 5.39i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 3.04iT - 89T^{2} \) |
| 97 | \( 1 + (-12.2 - 8.86i)T + (29.9 + 92.2i)T^{2} \) |
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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
−8.954530023226268974318557310571, −8.349459392637077194664202220907, −7.67312180480061487035191319666, −6.68170357303705193798599799540, −5.72385959817332687078964333069, −4.59249321428644806455080675297, −4.28278136786035970923910492781, −3.10777889546662267074955222092, −1.49928189067098936968398429232, −0.35379147676267765044238655773,
1.69279401703724002701278497399, 3.05340813240698837590500620276, 3.71660248940786038308417506591, 4.58126742112005814913301392013, 5.91046096265141088135157024888, 6.66717847671921746969236405347, 7.37012183836559341708440586683, 8.010613041333032875479062059743, 8.886550088476241812649360100325, 10.03281699411607521238326306863