| 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
−11.82986639234528253553141100108, −9.754718370064689367932310068641, −9.166806700324591631681220480419, −8.581183235775487967643954404911, −7.41047899182654446783784742024, −6.56975342826574862241151187286, −5.69107974083434486804357055355, −4.96845963360896971707730169242, −4.39520052924555951263562923899, −1.47798771182915634123274526996,
1.16515548646536077845784159361, 2.48716321741003830273174396910, 3.46585286488860410029395872989, 4.69942502136517776315640329223, 5.56395285511264060331754873476, 6.76013052818500761066411007456, 8.312765612126992531055570818501, 9.652789595983069141742063247670, 10.07780216222288131841293687278, 10.73761401571664318290968947304
