L(s) = 1 | + (−2.44 − 1.41i)5-s + (−1.73 + i)7-s + (1.41 + 2.44i)11-s + (1 − 1.73i)13-s + 6i·19-s + (2.82 − 4.89i)23-s + (1.49 + 2.59i)25-s + (−2.44 + 1.41i)29-s + (1.73 + i)31-s + 5.65·35-s + 6·37-s + (−4.89 − 2.82i)41-s + (−1.73 + i)43-s + (−5.65 − 9.79i)47-s + (−1.50 + 2.59i)49-s + ⋯ |
L(s) = 1 | + (−1.09 − 0.632i)5-s + (−0.654 + 0.377i)7-s + (0.426 + 0.738i)11-s + (0.277 − 0.480i)13-s + 1.37i·19-s + (0.589 − 1.02i)23-s + (0.299 + 0.519i)25-s + (−0.454 + 0.262i)29-s + (0.311 + 0.179i)31-s + 0.956·35-s + 0.986·37-s + (−0.765 − 0.441i)41-s + (−0.264 + 0.152i)43-s + (−0.825 − 1.42i)47-s + (−0.214 + 0.371i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8337837428\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8337837428\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.44 + 1.41i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.73 - i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.41 - 2.44i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 + 4.89i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.44 - 1.41i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.73 - i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + (4.89 + 2.82i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.73 - i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.65 + 9.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 8.48iT - 53T^{2} \) |
| 59 | \( 1 + (-1.41 + 2.44i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.73 - i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + (-12.1 + 7i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.41 + 2.44i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 16.9iT - 89T^{2} \) |
| 97 | \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.442948160145394938269625965323, −8.166552501078397720248133057716, −7.16104436514935613321023277450, −6.45733510777280653243713945757, −5.51754385178295863129829313625, −4.62746245858823980867003405688, −3.86189030360463292780502815222, −3.09912574663871891200449841805, −1.73199582666442101665696887790, −0.33897340861268764360857477232,
1.00672954437088836917923906201, 2.70637039956417355855745656734, 3.47631789447593650356972446553, 4.07359997140248211310547618920, 5.09550098753320275162422194318, 6.29415956247150383912411336699, 6.77325227873642803638586744188, 7.52522820637511313663205735863, 8.202282980923598241655904677045, 9.182778606058433410148846642288