| L(s) = 1 | + (0.492 + 1.35i)2-s + (0.333 − 1.69i)3-s + (−0.0577 + 0.0484i)4-s + (−0.440 − 2.49i)5-s + (2.46 − 0.386i)6-s + (−2.34 − 1.22i)7-s + (2.40 + 1.38i)8-s + (−2.77 − 1.13i)9-s + (3.16 − 1.82i)10-s + (2.06 + 0.364i)11-s + (0.0631 + 0.114i)12-s + (−0.0331 + 0.0909i)13-s + (0.494 − 3.77i)14-s + (−4.39 − 0.0834i)15-s + (−0.719 + 4.08i)16-s + (3.94 + 6.82i)17-s + ⋯ |
| L(s) = 1 | + (0.348 + 0.957i)2-s + (0.192 − 0.981i)3-s + (−0.0288 + 0.0242i)4-s + (−0.196 − 1.11i)5-s + (1.00 − 0.157i)6-s + (−0.887 − 0.461i)7-s + (0.848 + 0.490i)8-s + (−0.926 − 0.377i)9-s + (1.00 − 0.577i)10-s + (0.623 + 0.110i)11-s + (0.0182 + 0.0330i)12-s + (−0.00918 + 0.0252i)13-s + (0.132 − 1.01i)14-s + (−1.13 − 0.0215i)15-s + (−0.179 + 1.02i)16-s + (0.956 + 1.65i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44275 - 0.307492i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44275 - 0.307492i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.333 + 1.69i)T \) |
| 7 | \( 1 + (2.34 + 1.22i)T \) |
| good | 2 | \( 1 + (-0.492 - 1.35i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.440 + 2.49i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-2.06 - 0.364i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (0.0331 - 0.0909i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.94 - 6.82i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.53 + 0.887i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.82 - 4.56i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.573 - 1.57i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.916 + 1.09i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (1.88 + 3.26i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.60 + 0.946i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.03 + 5.87i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (3.23 + 2.71i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 9.25iT - 53T^{2} \) |
| 59 | \( 1 + (1.28 + 7.27i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (7.80 - 9.30i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (10.6 + 3.86i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (11.9 - 6.88i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-12.0 - 6.94i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.82 - 2.11i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-8.67 + 3.15i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (6.28 - 10.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.0 - 2.65i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.82468904078735030851083761770, −11.92453909534772781092083015461, −10.55529596583541491224390764805, −9.113352438512412208889522748574, −8.183886628346175020592141625186, −7.20621462899117643960022894666, −6.33223202097981006441558605048, −5.36139188535786929464737821170, −3.77551176489733535350703399594, −1.43443342096684117114878974243,
2.86806422372022048700118766663, 3.23355983414868432771983020173, 4.65505757168791267014211925499, 6.30838406301572199844220405953, 7.44299281271108580561744874844, 9.071881996459265916117308472653, 9.971118964103964335829238430336, 10.71502318541793298984821324996, 11.57342222627618342728977236780, 12.30954529262789813668995350127
