| L(s) = 1 | + (−1.34 + 1.05i)2-s + (0.814 − 0.580i)3-s + (0.221 − 0.912i)4-s + (−1.18 − 0.228i)5-s + (−0.482 + 1.64i)6-s + (0.252 − 2.63i)7-s + (−0.755 − 1.65i)8-s + (0.327 − 0.945i)9-s + (1.84 − 0.950i)10-s + (−2.19 + 2.79i)11-s + (−0.348 − 0.871i)12-s + (−3.01 + 4.68i)13-s + (2.45 + 3.81i)14-s + (−1.10 + 0.502i)15-s + (4.44 + 2.28i)16-s + (−4.10 + 3.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.952 + 0.749i)2-s + (0.470 − 0.334i)3-s + (0.110 − 0.456i)4-s + (−0.531 − 0.102i)5-s + (−0.197 + 0.671i)6-s + (0.0952 − 0.995i)7-s + (−0.267 − 0.585i)8-s + (0.109 − 0.315i)9-s + (0.582 − 0.300i)10-s + (−0.663 + 0.843i)11-s + (−0.100 − 0.251i)12-s + (−0.834 + 1.29i)13-s + (0.655 + 1.01i)14-s + (−0.284 + 0.129i)15-s + (1.11 + 0.572i)16-s + (−0.996 + 0.950i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0902990 + 0.415996i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0902990 + 0.415996i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.814 + 0.580i)T \) |
| 7 | \( 1 + (-0.252 + 2.63i)T \) |
| 23 | \( 1 + (1.33 - 4.60i)T \) |
| good | 2 | \( 1 + (1.34 - 1.05i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (1.18 + 0.228i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (2.19 - 2.79i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (3.01 - 4.68i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.10 - 3.91i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.09 - 4.85i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (-1.64 - 0.481i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.0541 - 0.567i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-0.0333 - 0.0115i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-3.96 - 3.43i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.02 - 0.467i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (2.39 - 1.38i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (9.31 - 0.443i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (3.53 + 6.86i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-8.72 + 12.2i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (2.47 - 6.17i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-0.277 - 1.92i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (2.50 + 0.608i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (11.3 + 0.541i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (2.67 + 3.08i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-5.86 + 0.560i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (4.96 - 5.73i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.28272062748232054539276819210, −9.934691602287355999850373585123, −9.627598003533050091875129901243, −8.397726190588586277481455427979, −7.65149451380838489322331879198, −7.27341308908244850540008750014, −6.29354599344700071961413139542, −4.55778989133421318391791328507, −3.62586431480925889915207342074, −1.71060403637186807131260545271,
0.32405459199681447300883070370, 2.54899064705260784139330195134, 2.96674413629285230273448112276, 4.86494889252347636646971497695, 5.69071088580653647050088167241, 7.42725689438304496739112457209, 8.211791711164387625758747561780, 8.937316738513061565434291645948, 9.647452836188624816084682644337, 10.54622656834475991800366246970
