| L(s) = 1 | + (−0.124 − 0.0875i)2-s + (1.70 + 0.288i)3-s + (−0.664 − 1.86i)4-s + (−3.83 + 2.12i)5-s + (−0.186 − 0.185i)6-s + (1.98 − 1.00i)7-s + (−0.161 + 0.582i)8-s + (2.83 + 0.984i)9-s + (0.661 + 0.0719i)10-s + (5.63 − 2.12i)11-s + (−0.598 − 3.37i)12-s + (−0.494 − 3.89i)13-s + (−0.334 − 0.0485i)14-s + (−7.15 + 2.51i)15-s + (−2.98 + 2.44i)16-s + (0.109 + 2.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.0878 − 0.0618i)2-s + (0.986 + 0.166i)3-s + (−0.332 − 0.930i)4-s + (−1.71 + 0.948i)5-s + (−0.0762 − 0.0756i)6-s + (0.749 − 0.379i)7-s + (−0.0571 + 0.205i)8-s + (0.944 + 0.328i)9-s + (0.209 + 0.0227i)10-s + (1.69 − 0.641i)11-s + (−0.172 − 0.973i)12-s + (−0.137 − 1.07i)13-s + (−0.0892 − 0.0129i)14-s + (−1.84 + 0.650i)15-s + (−0.747 + 0.611i)16-s + (0.0265 + 0.488i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48490 - 0.497606i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48490 - 0.497606i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.70 - 0.288i)T \) |
| 59 | \( 1 + (7.47 - 1.77i)T \) |
| good | 2 | \( 1 + (0.124 + 0.0875i)T + (0.672 + 1.88i)T^{2} \) |
| 5 | \( 1 + (3.83 - 2.12i)T + (2.65 - 4.23i)T^{2} \) |
| 7 | \( 1 + (-1.98 + 1.00i)T + (4.13 - 5.64i)T^{2} \) |
| 11 | \( 1 + (-5.63 + 2.12i)T + (8.25 - 7.27i)T^{2} \) |
| 13 | \( 1 + (0.494 + 3.89i)T + (-12.5 + 3.25i)T^{2} \) |
| 17 | \( 1 + (-0.109 - 2.01i)T + (-16.9 + 1.83i)T^{2} \) |
| 19 | \( 1 + (-2.84 + 2.69i)T + (1.02 - 18.9i)T^{2} \) |
| 23 | \( 1 + (-0.784 + 2.47i)T + (-18.8 - 13.2i)T^{2} \) |
| 29 | \( 1 + (-0.0501 - 0.553i)T + (-28.5 + 5.20i)T^{2} \) |
| 31 | \( 1 + (-2.08 + 0.621i)T + (25.9 - 16.9i)T^{2} \) |
| 37 | \( 1 + (-1.06 - 3.83i)T + (-31.7 + 19.0i)T^{2} \) |
| 41 | \( 1 + (-2.92 + 3.19i)T + (-3.69 - 40.8i)T^{2} \) |
| 43 | \( 1 + (3.05 - 2.50i)T + (8.48 - 42.1i)T^{2} \) |
| 47 | \( 1 + (1.12 + 0.622i)T + (24.9 + 39.8i)T^{2} \) |
| 53 | \( 1 + (-3.72 + 0.404i)T + (51.7 - 11.3i)T^{2} \) |
| 61 | \( 1 + (11.1 + 7.89i)T + (20.5 + 57.4i)T^{2} \) |
| 67 | \( 1 + (-5.92 - 6.03i)T + (-1.20 + 66.9i)T^{2} \) |
| 71 | \( 1 + (8.86 - 5.33i)T + (33.2 - 62.7i)T^{2} \) |
| 73 | \( 1 + (-0.0986 - 0.247i)T + (-52.9 + 50.2i)T^{2} \) |
| 79 | \( 1 + (-6.02 - 5.30i)T + (9.95 + 78.3i)T^{2} \) |
| 83 | \( 1 + (15.0 + 1.09i)T + (82.1 + 11.9i)T^{2} \) |
| 89 | \( 1 + (-6.15 - 2.84i)T + (57.6 + 67.8i)T^{2} \) |
| 97 | \( 1 + (-5.67 + 0.826i)T + (92.9 - 27.6i)T^{2} \) |
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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
−10.84354872866614774226099925248, −9.936068497197728758378939872964, −8.803318701701470152735957588268, −8.173376940105388205147629054101, −7.31956127827334373804728979292, −6.39480520108612510147473022726, −4.69577861338922379741829879985, −3.93922650161537825102035670875, −2.99638394598369565282308587365, −1.05904064118792693637721820565,
1.49113090678192543694160460331, 3.37007270913124498747799483360, 4.19343234241195942253763308288, 4.66819084418365524612938956915, 6.95196561884278833805314528022, 7.58636110081583849731027114490, 8.243758115060488190834312788711, 9.111165801764880990094881384727, 9.327823912228964649691990846563, 11.45978533075476136086517195328
