| L(s) = 1 | + (−1.72 − 0.197i)2-s + (−2.49 − 1.80i)3-s + (0.982 + 0.228i)4-s + (−0.564 + 0.825i)5-s + (3.93 + 3.60i)6-s + (−0.450 − 0.463i)7-s + (1.61 + 0.577i)8-s + (2.00 + 6.15i)9-s + (1.13 − 1.31i)10-s + (−3.30 + 0.228i)11-s + (−2.03 − 2.34i)12-s + (3.16 − 5.24i)13-s + (0.685 + 0.888i)14-s + (2.89 − 1.03i)15-s + (−4.48 − 2.20i)16-s + (5.35 − 4.37i)17-s + ⋯ |
| L(s) = 1 | + (−1.21 − 0.139i)2-s + (−1.43 − 1.04i)3-s + (0.491 + 0.114i)4-s + (−0.252 + 0.369i)5-s + (1.60 + 1.47i)6-s + (−0.170 − 0.175i)7-s + (0.572 + 0.204i)8-s + (0.666 + 2.05i)9-s + (0.359 − 0.414i)10-s + (−0.997 + 0.0688i)11-s + (−0.586 − 0.677i)12-s + (0.877 − 1.45i)13-s + (0.183 + 0.237i)14-s + (0.748 − 0.267i)15-s + (−1.12 − 0.551i)16-s + (1.29 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0353199 - 0.262317i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0353199 - 0.262317i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.564 - 0.825i)T \) |
| 11 | \( 1 + (3.30 - 0.228i)T \) |
| good | 2 | \( 1 + (1.72 + 0.197i)T + (1.94 + 0.452i)T^{2} \) |
| 3 | \( 1 + (2.49 + 1.80i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (0.450 + 0.463i)T + (-0.199 + 6.99i)T^{2} \) |
| 13 | \( 1 + (-3.16 + 5.24i)T + (-6.06 - 11.4i)T^{2} \) |
| 17 | \( 1 + (-5.35 + 4.37i)T + (3.37 - 16.6i)T^{2} \) |
| 19 | \( 1 + (-6.34 - 0.363i)T + (18.8 + 2.16i)T^{2} \) |
| 23 | \( 1 + (0.520 - 3.61i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + (1.60 - 4.11i)T + (-21.3 - 19.6i)T^{2} \) |
| 31 | \( 1 + (2.52 + 1.06i)T + (21.6 + 22.2i)T^{2} \) |
| 37 | \( 1 + (-0.144 + 1.67i)T + (-36.4 - 6.30i)T^{2} \) |
| 41 | \( 1 + (-7.86 + 4.44i)T + (21.1 - 35.1i)T^{2} \) |
| 43 | \( 1 + (11.4 - 3.36i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-0.567 + 2.80i)T + (-43.2 - 18.2i)T^{2} \) |
| 53 | \( 1 + (8.37 - 4.11i)T + (32.3 - 41.9i)T^{2} \) |
| 59 | \( 1 + (-2.88 - 1.62i)T + (30.4 + 50.5i)T^{2} \) |
| 61 | \( 1 + (-3.61 + 0.414i)T + (59.4 - 13.8i)T^{2} \) |
| 67 | \( 1 + (-1.87 + 4.10i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.443 - 1.68i)T + (-61.8 + 34.9i)T^{2} \) |
| 73 | \( 1 + (3.58 + 6.78i)T + (-41.2 + 60.2i)T^{2} \) |
| 79 | \( 1 + (-0.248 - 8.70i)T + (-78.8 + 4.51i)T^{2} \) |
| 83 | \( 1 + (9.88 + 1.71i)T + (78.1 + 27.8i)T^{2} \) |
| 89 | \( 1 + (3.61 + 2.32i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (3.91 + 5.72i)T + (-35.1 + 90.3i)T^{2} \) |
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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
−10.33855579854377238157916570851, −9.684482125363308743583833896349, −8.142272320768992050538636010284, −7.57438007938095702079871801212, −7.09269915964888857360885660192, −5.59716810321596891025589215703, −5.26412324983554857973743731875, −3.11715385364235868018125546433, −1.34476528655644767461681966123, −0.34861730911771003786057888783,
1.19384890008440775779100391695, 3.71666548204249571083679769908, 4.65930652774101619720410880915, 5.62286798154057848021404665491, 6.51390648516109904055601636681, 7.70505751027478206432244246330, 8.616932156857180168380110036724, 9.604585208532555580936765545152, 9.988940592567915877109054350306, 10.91422025140246335952560694905
