| L(s) = 1 | + (−2.15 − 1.03i)2-s + (2.31 + 2.90i)4-s + (3.04 − 2.82i)5-s + (−1.76 − 1.01i)7-s + (−0.911 − 3.99i)8-s + (−9.49 + 2.92i)10-s + (−3.31 − 2.64i)11-s + (4.78 + 1.47i)13-s + (2.74 + 4.02i)14-s + (−0.526 + 2.30i)16-s + (2.10 − 2.26i)17-s + (0.730 + 0.286i)19-s + (15.2 + 2.30i)20-s + (4.39 + 9.13i)22-s + (−0.641 + 4.25i)23-s + ⋯ |
| L(s) = 1 | + (−1.52 − 0.733i)2-s + (1.15 + 1.45i)4-s + (1.36 − 1.26i)5-s + (−0.666 − 0.384i)7-s + (−0.322 − 1.41i)8-s + (−3.00 + 0.926i)10-s + (−0.999 − 0.797i)11-s + (1.32 + 0.409i)13-s + (0.732 + 1.07i)14-s + (−0.131 + 0.576i)16-s + (0.510 − 0.549i)17-s + (0.167 + 0.0657i)19-s + (3.41 + 0.514i)20-s + (0.937 + 1.94i)22-s + (−0.133 + 0.887i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.262935 - 0.651149i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.262935 - 0.651149i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (-0.170 - 6.55i)T \) |
| good | 2 | \( 1 + (2.15 + 1.03i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (-3.04 + 2.82i)T + (0.373 - 4.98i)T^{2} \) |
| 7 | \( 1 + (1.76 + 1.01i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.31 + 2.64i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-4.78 - 1.47i)T + (10.7 + 7.32i)T^{2} \) |
| 17 | \( 1 + (-2.10 + 2.26i)T + (-1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.730 - 0.286i)T + (13.9 + 12.9i)T^{2} \) |
| 23 | \( 1 + (0.641 - 4.25i)T + (-21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (4.18 - 2.85i)T + (10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (0.630 + 8.40i)T + (-30.6 + 4.62i)T^{2} \) |
| 37 | \( 1 + (1.83 - 1.06i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.36 + 6.98i)T + (-25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (4.61 - 3.68i)T + (10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (2.59 + 8.41i)T + (-43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (4.46 + 1.01i)T + (53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (0.719 + 0.0539i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-3.21 + 8.18i)T + (-49.1 - 45.5i)T^{2} \) |
| 71 | \( 1 + (-7.32 + 1.10i)T + (67.8 - 20.9i)T^{2} \) |
| 73 | \( 1 + (4.31 - 13.9i)T + (-60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (-1.58 + 2.74i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.42 + 6.49i)T + (-30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (-7.63 - 5.20i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (-0.938 + 1.17i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−10.74335319656812922765351359683, −9.766244132290696189070699563893, −9.392554849179344551680213012979, −8.554034998371770961133894652372, −7.72372882126909530575664468316, −6.22557948363434067152144649174, −5.30566100589414180632702429428, −3.40547896504200793946335252698, −1.95207534331160221430249228152, −0.77504994720809036072920404895,
1.81372556921168940466924529912, 3.07222119578389978463834861445, 5.61034858913851454019422861984, 6.23524872085965776836927944126, 6.97826537353072803598045665869, 7.977846460264524602501909073597, 9.036227620961720818759883904460, 9.835229660815661423307674562676, 10.45764339975889189328595764333, 10.89187696666799329070057969299
