| L(s) = 1 | + (0.385 − 2.68i)2-s + (−0.909 + 0.415i)3-s + (−5.11 − 1.50i)4-s + (2.40 + 2.77i)5-s + (0.762 + 2.59i)6-s + (2.27 − 1.34i)7-s + (−3.74 + 8.20i)8-s + (0.654 − 0.755i)9-s + (8.37 − 5.38i)10-s + (3.87 − 0.557i)11-s + (5.27 − 0.758i)12-s + (−3.02 − 4.70i)13-s + (−2.73 − 6.62i)14-s + (−3.34 − 1.52i)15-s + (11.5 + 7.44i)16-s + (−0.384 + 0.112i)17-s + ⋯ |
| L(s) = 1 | + (0.272 − 1.89i)2-s + (−0.525 + 0.239i)3-s + (−2.55 − 0.751i)4-s + (1.07 + 1.24i)5-s + (0.311 + 1.06i)6-s + (0.860 − 0.509i)7-s + (−1.32 + 2.90i)8-s + (0.218 − 0.251i)9-s + (2.64 − 1.70i)10-s + (1.16 − 0.168i)11-s + (1.52 − 0.219i)12-s + (−0.839 − 1.30i)13-s + (−0.731 − 1.76i)14-s + (−0.863 − 0.394i)15-s + (2.89 + 1.86i)16-s + (−0.0932 + 0.0273i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.712069 - 1.36280i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.712069 - 1.36280i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.909 - 0.415i)T \) |
| 7 | \( 1 + (-2.27 + 1.34i)T \) |
| 23 | \( 1 + (-1.28 + 4.61i)T \) |
| good | 2 | \( 1 + (-0.385 + 2.68i)T + (-1.91 - 0.563i)T^{2} \) |
| 5 | \( 1 + (-2.40 - 2.77i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-3.87 + 0.557i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (3.02 + 4.70i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.384 - 0.112i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-3.23 - 0.950i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-3.22 + 0.947i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.10 - 1.41i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.641 + 0.555i)T + (5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-1.20 + 1.04i)T + (5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (9.48 - 4.33i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 2.93iT - 47T^{2} \) |
| 53 | \( 1 + (-3.96 + 6.17i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (4.63 + 7.21i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (0.480 - 1.05i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-4.59 - 0.661i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (0.365 - 2.53i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (1.83 - 6.26i)T + (-61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-3.42 - 5.33i)T + (-32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (10.6 - 12.2i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (0.0437 + 0.0957i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (10.0 + 11.5i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.73761401571664318290968947304, −10.07780216222288131841293687278, −9.652789595983069141742063247670, −8.312765612126992531055570818501, −6.76013052818500761066411007456, −5.56395285511264060331754873476, −4.69942502136517776315640329223, −3.46585286488860410029395872989, −2.48716321741003830273174396910, −1.16515548646536077845784159361,
1.47798771182915634123274526996, 4.39520052924555951263562923899, 4.96845963360896971707730169242, 5.69107974083434486804357055355, 6.56975342826574862241151187286, 7.41047899182654446783784742024, 8.581183235775487967643954404911, 9.166806700324591631681220480419, 9.754718370064689367932310068641, 11.82986639234528253553141100108
