L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (−0.635 − 1.39i)5-s + (−0.142 + 0.989i)6-s + (0.959 − 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (−1.46 − 0.430i)10-s + (0.0123 + 0.0142i)11-s + (0.654 + 0.755i)12-s + (−3.03 − 0.891i)13-s + (0.415 − 0.909i)14-s + (1.28 + 0.826i)15-s + (−0.959 + 0.281i)16-s + (0.240 − 1.66i)17-s + ⋯ |
L(s) = 1 | + (0.463 − 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.284 − 0.622i)5-s + (−0.0580 + 0.404i)6-s + (0.362 − 0.106i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.464 − 0.136i)10-s + (0.00372 + 0.00430i)11-s + (0.189 + 0.218i)12-s + (−0.841 − 0.247i)13-s + (0.111 − 0.243i)14-s + (0.332 + 0.213i)15-s + (−0.239 + 0.0704i)16-s + (0.0582 − 0.404i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.133118 - 0.942213i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.133118 - 0.942213i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 3 | \( 1 + (0.841 - 0.540i)T \) |
| 7 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (4.34 - 2.02i)T \) |
good | 5 | \( 1 + (0.635 + 1.39i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (-0.0123 - 0.0142i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (3.03 + 0.891i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.240 + 1.66i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.432 + 3.01i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.688 - 4.78i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (6.68 + 4.29i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-3.07 + 6.72i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (2.20 + 4.83i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (6.27 - 4.03i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 0.302T + 47T^{2} \) |
| 53 | \( 1 + (7.87 - 2.31i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (6.28 + 1.84i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (0.145 + 0.0932i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-3.68 + 4.24i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (0.505 - 0.583i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.36 + 9.51i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-7.89 - 2.31i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (0.812 - 1.77i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-7.68 + 4.93i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (1.73 + 3.79i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−9.685954805968949117812496076340, −9.091641106457106333918735499209, −7.956169310075601547169717703816, −7.06591618072328511886586953510, −5.87586273749261631457937105542, −5.01832199116474674135178819786, −4.45863321373273528917014591770, −3.35681480116263103668846400219, −1.96326326948068528517548219142, −0.38610916890690501338301124290,
1.91469276432351671475950241479, 3.25574539178615857294990781708, 4.36879255106762216253600507443, 5.26463031372963295721498168247, 6.20394662019028160853389160315, 6.91396550393516287911437241701, 7.73389911933978775740363264576, 8.381591343534806292942564353341, 9.643214267903530296196549703490, 10.49106578354601666682530647490