L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (2.62 + 1.51i)5-s − 0.999i·6-s + (2.43 − 1.04i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1.51 − 2.62i)10-s + (3.75 − 2.16i)11-s + (−0.499 + 0.866i)12-s + (−0.613 − 3.55i)13-s + (−2.62 − 0.311i)14-s + 3.02i·15-s + (−0.5 + 0.866i)16-s + (−0.721 − 1.25i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (1.17 + 0.676i)5-s − 0.408i·6-s + (0.918 − 0.394i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.478 − 0.829i)10-s + (1.13 − 0.653i)11-s + (−0.144 + 0.249i)12-s + (−0.170 − 0.985i)13-s + (−0.702 − 0.0831i)14-s + 0.781i·15-s + (−0.125 + 0.216i)16-s + (−0.175 − 0.303i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.62358 + 0.0946433i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.62358 + 0.0946433i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.43 + 1.04i)T \) |
| 13 | \( 1 + (0.613 + 3.55i)T \) |
good | 5 | \( 1 + (-2.62 - 1.51i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.75 + 2.16i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (0.721 + 1.25i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.12 + 0.652i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.36 - 4.09i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.42T + 29T^{2} \) |
| 31 | \( 1 + (1.46 - 0.847i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (9.89 + 5.71i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.53iT - 41T^{2} \) |
| 43 | \( 1 + 5.10T + 43T^{2} \) |
| 47 | \( 1 + (-7.28 - 4.20i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.05 - 3.55i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.34 - 2.50i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.505 - 0.875i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.412 - 0.238i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12.4iT - 71T^{2} \) |
| 73 | \( 1 + (-3.42 + 1.97i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.34 - 12.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 17.0iT - 83T^{2} \) |
| 89 | \( 1 + (6.04 + 3.48i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 15.9iT - 97T^{2} \) |
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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
−10.67690136887016786371211641441, −10.00346618028256843841300432915, −9.195393340642317290656572398414, −8.378273190003250996803411270657, −7.36423498104698312915391910199, −6.29562400325052327354302221524, −5.26049102870030624238794636799, −3.86056257085030624195789561496, −2.73388509126172457627369247109, −1.47946837832586067353110970447,
1.56315060232583956956166894545, 2.07271458455986985916146775927, 4.32185577163103912425546168307, 5.38065433926114441705173067348, 6.40278314927122059622499992687, 7.10525044739723835198583494356, 8.475662219865134465280059265819, 8.828371974463917002297331341674, 9.645961945600721796363419894168, 10.54001952005028192398212562514