| L(s) = 1 | + (0.766 + 0.642i)2-s + (2.57 − 0.936i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (2.57 + 0.936i)6-s + (−1.92 − 3.33i)7-s + (−0.500 + 0.866i)8-s + (3.44 − 2.89i)9-s + (−0.766 + 0.642i)10-s + (−2.86 + 4.96i)11-s + (1.36 + 2.37i)12-s + (−5.05 − 1.84i)13-s + (0.668 − 3.79i)14-s + (0.475 + 2.69i)15-s + (−0.939 + 0.342i)16-s + (1.05 + 0.883i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (1.48 − 0.540i)3-s + (0.0868 + 0.492i)4-s + (−0.0776 + 0.440i)5-s + (1.05 + 0.382i)6-s + (−0.727 − 1.25i)7-s + (−0.176 + 0.306i)8-s + (1.14 − 0.964i)9-s + (−0.242 + 0.203i)10-s + (−0.863 + 1.49i)11-s + (0.395 + 0.684i)12-s + (−1.40 − 0.510i)13-s + (0.178 − 1.01i)14-s + (0.122 + 0.696i)15-s + (−0.234 + 0.0855i)16-s + (0.255 + 0.214i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.04857 + 0.263105i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04857 + 0.263105i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-4.23 + 1.03i)T \) |
| good | 3 | \( 1 + (-2.57 + 0.936i)T + (2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (1.92 + 3.33i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.86 - 4.96i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.05 + 1.84i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.05 - 0.883i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.274 - 1.55i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.22 + 4.38i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.07 - 5.32i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.89T + 37T^{2} \) |
| 41 | \( 1 + (3.39 - 1.23i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.568 + 3.22i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.55 + 4.65i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (2.04 + 11.5i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (2.00 + 1.68i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.05 + 5.99i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (4.81 - 4.03i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.16 - 12.2i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-8.85 + 3.22i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (7.27 - 2.64i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.24 - 3.88i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.964 - 0.351i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (1.01 + 0.849i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.91512350161872759113472002905, −12.15595046058977429456468358383, −10.19201435995441100247614678706, −9.762070364469574059080911162078, −8.111492708308547600269018894760, −7.28855016938681370664397862977, −6.98375017761522378523393844247, −4.88172234831378818621011698338, −3.49840019397184608723624620879, −2.49591536989272793105468949595,
2.61173551871950738478164492101, 3.16293984811947091842763952727, 4.74928750618917966066381272677, 5.87829154359292339763762019920, 7.71231576883375862529409504161, 8.808305580474435983001340365744, 9.398670727567954792602488191769, 10.32853379874434223226245533513, 11.82299765402576832021296400426, 12.57814593222667578899476723755
