| L(s) = 1 | + (0.00892 + 0.00102i)2-s + (−0.467 − 0.339i)3-s + (−1.94 − 0.452i)4-s + (−0.564 + 0.825i)5-s + (−0.00382 − 0.00350i)6-s + (1.06 + 1.09i)7-s + (−0.0338 − 0.0120i)8-s + (−0.823 − 2.53i)9-s + (−0.00588 + 0.00678i)10-s + (−0.898 + 3.19i)11-s + (0.756 + 0.873i)12-s + (0.398 − 0.661i)13-s + (0.00834 + 0.0108i)14-s + (0.544 − 0.194i)15-s + (3.58 + 1.76i)16-s + (3.12 − 2.55i)17-s + ⋯ |
| L(s) = 1 | + (0.00630 + 0.000723i)2-s + (−0.269 − 0.196i)3-s + (−0.973 − 0.226i)4-s + (−0.252 + 0.369i)5-s + (−0.00156 − 0.00143i)6-s + (0.400 + 0.412i)7-s + (−0.0119 − 0.00426i)8-s + (−0.274 − 0.845i)9-s + (−0.00186 + 0.00214i)10-s + (−0.270 + 0.962i)11-s + (0.218 + 0.252i)12-s + (0.110 − 0.183i)13-s + (0.00222 + 0.00289i)14-s + (0.140 − 0.0501i)15-s + (0.897 + 0.441i)16-s + (0.758 − 0.620i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01287 - 0.215363i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01287 - 0.215363i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.898 - 3.19i)T \) |
| good | 2 | \( 1 + (-0.00892 - 0.00102i)T + (1.94 + 0.452i)T^{2} \) |
| 3 | \( 1 + (0.467 + 0.339i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (-1.06 - 1.09i)T + (-0.199 + 6.99i)T^{2} \) |
| 13 | \( 1 + (-0.398 + 0.661i)T + (-6.06 - 11.4i)T^{2} \) |
| 17 | \( 1 + (-3.12 + 2.55i)T + (3.37 - 16.6i)T^{2} \) |
| 19 | \( 1 + (-5.99 - 0.342i)T + (18.8 + 2.16i)T^{2} \) |
| 23 | \( 1 + (-0.897 + 6.24i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + (-1.00 + 2.59i)T + (-21.3 - 19.6i)T^{2} \) |
| 31 | \( 1 + (-1.31 - 0.556i)T + (21.6 + 22.2i)T^{2} \) |
| 37 | \( 1 + (0.326 - 3.79i)T + (-36.4 - 6.30i)T^{2} \) |
| 41 | \( 1 + (-9.68 + 5.46i)T + (21.1 - 35.1i)T^{2} \) |
| 43 | \( 1 + (-6.28 + 1.84i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (1.00 - 4.95i)T + (-43.2 - 18.2i)T^{2} \) |
| 53 | \( 1 + (-4.04 + 1.98i)T + (32.3 - 41.9i)T^{2} \) |
| 59 | \( 1 + (-6.16 - 3.48i)T + (30.4 + 50.5i)T^{2} \) |
| 61 | \( 1 + (4.62 - 0.531i)T + (59.4 - 13.8i)T^{2} \) |
| 67 | \( 1 + (2.24 - 4.91i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (1.37 + 5.21i)T + (-61.8 + 34.9i)T^{2} \) |
| 73 | \( 1 + (-7.44 - 14.1i)T + (-41.2 + 60.2i)T^{2} \) |
| 79 | \( 1 + (0.193 + 6.76i)T + (-78.8 + 4.51i)T^{2} \) |
| 83 | \( 1 + (1.15 + 0.199i)T + (78.1 + 27.8i)T^{2} \) |
| 89 | \( 1 + (9.75 + 6.27i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.20 - 3.22i)T + (-35.1 + 90.3i)T^{2} \) |
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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
−10.45434579903599730544120539604, −9.695162566891397701667797603871, −8.952806312660684270417285373906, −7.956033197934808885793321430145, −7.08908172288845194239997238598, −5.88122409181478795580211903407, −5.10903754937814733690702275609, −4.06280824210515560772827175737, −2.79564390183125257238240885734, −0.856173505411562630880798860881,
1.04639305738823226523500949768, 3.18989524377805811299777352858, 4.18791577828943494900041580955, 5.21692896747418802072984262869, 5.74496701359316840510503424721, 7.62026809637234453331292781542, 7.954506943881099268415214829720, 8.961223985361898498575472219810, 9.774390963637351813364647391564, 10.79402935322465667929944782550
