| L(s) = 1 | + (0.471 − 1.29i)2-s + (1.37 − 1.05i)3-s + (0.0766 + 0.0643i)4-s + (−0.459 + 2.60i)5-s + (−0.710 − 2.27i)6-s + (−1.47 − 2.19i)7-s + (2.50 − 1.44i)8-s + (0.794 − 2.89i)9-s + (3.15 + 1.82i)10-s + (−0.681 + 0.120i)11-s + (0.173 + 0.00811i)12-s + (0.171 + 0.472i)13-s + (−3.54 + 0.869i)14-s + (2.10 + 4.06i)15-s + (−0.658 − 3.73i)16-s + (−2.42 + 4.19i)17-s + ⋯ |
| L(s) = 1 | + (0.333 − 0.915i)2-s + (0.795 − 0.606i)3-s + (0.0383 + 0.0321i)4-s + (−0.205 + 1.16i)5-s + (−0.290 − 0.930i)6-s + (−0.556 − 0.831i)7-s + (0.886 − 0.511i)8-s + (0.264 − 0.964i)9-s + (0.998 + 0.576i)10-s + (−0.205 + 0.0362i)11-s + (0.0499 + 0.00234i)12-s + (0.0476 + 0.130i)13-s + (−0.946 + 0.232i)14-s + (0.542 + 1.05i)15-s + (−0.164 − 0.933i)16-s + (−0.587 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42553 - 1.01530i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42553 - 1.01530i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.37 + 1.05i)T \) |
| 7 | \( 1 + (1.47 + 2.19i)T \) |
| good | 2 | \( 1 + (-0.471 + 1.29i)T + (-1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (0.459 - 2.60i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (0.681 - 0.120i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.171 - 0.472i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.42 - 4.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.03 - 0.597i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.74 - 5.65i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (2.19 - 6.03i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.17 + 6.16i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (3.71 - 6.43i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.25 + 1.18i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.81 + 10.2i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (2.93 - 2.45i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (-1.13 + 6.41i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.36 - 2.81i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.24 - 1.17i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.286 - 0.165i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.13 + 1.81i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.71 - 3.53i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-12.8 - 4.69i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (4.84 + 8.39i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.19 - 0.563i)T + (91.1 - 33.1i)T^{2} \) |
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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
−12.45365791844966135882962770797, −11.42036386186580446990480665554, −10.52712646230545063771119539118, −9.769274357327895183001010544789, −8.124402493022341200202868924940, −7.18135265385879232564925689714, −6.48415473079945251517217705821, −3.92681914747331811243428291756, −3.28143965785202089959600181651, −1.96629155887473442091397286521,
2.46473636057687317526411836033, 4.38449105016546942929354790308, 5.18554970005878221483771185525, 6.40116141094811099604212992663, 7.83911499354824572450358577920, 8.622178579965794126530868732440, 9.465579788540658818933861463518, 10.63258752905383182629109451137, 11.99622050002741737046751947687, 13.04368258661786110407546908687
