| L(s) = 1 | + (−0.766 − 0.642i)2-s + (2.22 − 0.811i)3-s + (0.173 + 0.984i)4-s + (0.173 − 0.984i)5-s + (−2.22 − 0.811i)6-s + (−1.57 − 2.73i)7-s + (0.500 − 0.866i)8-s + (2.00 − 1.68i)9-s + (−0.766 + 0.642i)10-s + (−0.688 + 1.19i)11-s + (1.18 + 2.05i)12-s + (4.06 + 1.47i)13-s + (−0.547 + 3.10i)14-s + (−0.411 − 2.33i)15-s + (−0.939 + 0.342i)16-s + (−0.993 − 0.833i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (1.28 − 0.468i)3-s + (0.0868 + 0.492i)4-s + (0.0776 − 0.440i)5-s + (−0.909 − 0.331i)6-s + (−0.596 − 1.03i)7-s + (0.176 − 0.306i)8-s + (0.669 − 0.562i)9-s + (−0.242 + 0.203i)10-s + (−0.207 + 0.359i)11-s + (0.342 + 0.592i)12-s + (1.12 + 0.410i)13-s + (−0.146 + 0.830i)14-s + (−0.106 − 0.602i)15-s + (−0.234 + 0.0855i)16-s + (−0.240 − 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02308 - 0.780271i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02308 - 0.780271i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 + (-1.08 - 4.22i)T \) |
| good | 3 | \( 1 + (-2.22 + 0.811i)T + (2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (1.57 + 2.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.688 - 1.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.06 - 1.47i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.993 + 0.833i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.369 + 2.09i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.0998 - 0.0837i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.173 + 0.300i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + (10.3 - 3.76i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (2.00 - 11.3i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.0585 + 0.0491i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.08 - 6.12i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (6.27 + 5.26i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.36 - 7.72i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.789 - 0.662i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.02 + 11.4i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-11.5 + 4.18i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (13.1 - 4.79i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (5.68 + 9.85i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-17.1 - 6.23i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (12.4 + 10.4i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.60545768587124181541100750354, −11.28844509473031360665311263462, −10.11575281475099341648920487182, −9.330014769711474431157624986114, −8.337628661285927640435933743737, −7.61722741107381077511211732274, −6.43987302792540986678398717073, −4.20830825846434393385191762423, −3.09797616203591873879154278088, −1.49522190374989653156568944878,
2.49896957647338591792666511834, 3.58785302087501228870285377368, 5.53406177719607673492853812023, 6.64637133615044384339441840193, 8.042757541769009150680843746978, 8.795454116897007075933615901930, 9.442069683797859126627549771155, 10.46844417170843434428926566329, 11.58414137219503226170153013407, 13.14482218625273110516745711319
