L(s) = 1 | + (0.730 + 1.26i)2-s + (−0.0665 + 0.115i)4-s + (0.296 + 0.514i)5-s + (2.32 − 1.26i)7-s + 2.72·8-s + (−0.433 + 0.750i)10-s + (−2.23 + 3.86i)11-s − 4.51·13-s + (3.29 + 2.01i)14-s + (2.12 + 3.67i)16-s + (0.136 − 0.236i)17-s + (−1.43 − 2.48i)19-s − 0.0789·20-s − 6.51·22-s + (−2.52 − 4.37i)23-s + ⋯ |
L(s) = 1 | + (0.516 + 0.894i)2-s + (−0.0332 + 0.0576i)4-s + (0.132 + 0.229i)5-s + (0.878 − 0.478i)7-s + 0.964·8-s + (−0.137 + 0.237i)10-s + (−0.672 + 1.16i)11-s − 1.25·13-s + (0.881 + 0.538i)14-s + (0.531 + 0.919i)16-s + (0.0331 − 0.0574i)17-s + (−0.328 − 0.569i)19-s − 0.0176·20-s − 1.38·22-s + (−0.526 − 0.912i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48105 + 0.769802i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48105 + 0.769802i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.32 + 1.26i)T \) |
good | 2 | \( 1 + (-0.730 - 1.26i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-0.296 - 0.514i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.23 - 3.86i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.51T + 13T^{2} \) |
| 17 | \( 1 + (-0.136 + 0.236i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.43 + 2.48i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.52 + 4.37i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 0.352T + 29T^{2} \) |
| 31 | \( 1 + (1.25 - 2.17i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.32 - 5.75i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 3.38T + 43T^{2} \) |
| 47 | \( 1 + (6.21 + 10.7i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.66 - 9.80i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.02 + 6.97i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.36 - 2.36i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.93 - 5.08i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.60T + 71T^{2} \) |
| 73 | \( 1 + (5.55 - 9.62i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.58 + 9.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 + (-2.68 - 4.65i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2.26T + 97T^{2} \) |
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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
−12.93969963374284468405716770875, −11.87092423899478930050305658426, −10.53595193207495602594035410119, −10.03177039228435817918511922381, −8.271887336887227106650469457814, −7.34614281512022805531858965714, −6.62151194074563675045925169339, −5.04616122491773614662085498031, −4.56370563194415387173852405404, −2.18925864776931347484174428161,
1.92959202178053118598540503962, 3.24238314136025927764919275467, 4.72735538671741628648836767723, 5.65753152070042961390964534084, 7.51102597759450757616874456943, 8.302962994162382241992868589100, 9.666873093922442376984319968456, 10.82088478441631392524790274220, 11.49420622459907807941323074091, 12.35286126164403648332282772689