L(s) = 1 | + (−1.68 − 0.412i)3-s + (−0.379 − 0.219i)5-s + (2.14 + 3.71i)7-s + (2.65 + 1.38i)9-s + (3.41 − 1.97i)11-s + (−3.44 − 1.98i)13-s + (0.548 + 0.525i)15-s − 1.25·17-s + 6.44i·19-s + (−2.07 − 7.13i)21-s + (−2.32 + 4.02i)23-s + (−2.40 − 4.16i)25-s + (−3.90 − 3.43i)27-s + (5.46 − 3.15i)29-s + (−0.910 + 1.57i)31-s + ⋯ |
L(s) = 1 | + (−0.971 − 0.238i)3-s + (−0.169 − 0.0980i)5-s + (0.810 + 1.40i)7-s + (0.886 + 0.462i)9-s + (1.03 − 0.594i)11-s + (−0.954 − 0.551i)13-s + (0.141 + 0.135i)15-s − 0.304·17-s + 1.47i·19-s + (−0.452 − 1.55i)21-s + (−0.484 + 0.839i)23-s + (−0.480 − 0.832i)25-s + (−0.750 − 0.660i)27-s + (1.01 − 0.585i)29-s + (−0.163 + 0.283i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.070379093\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.070379093\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.68 + 0.412i)T \) |
good | 5 | \( 1 + (0.379 + 0.219i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.14 - 3.71i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.41 + 1.97i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.44 + 1.98i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.25T + 17T^{2} \) |
| 19 | \( 1 - 6.44iT - 19T^{2} \) |
| 23 | \( 1 + (2.32 - 4.02i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.46 + 3.15i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.910 - 1.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.87iT - 37T^{2} \) |
| 41 | \( 1 + (-1.53 + 2.65i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.45 + 0.838i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.01 - 8.68i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 11.4iT - 53T^{2} \) |
| 59 | \( 1 + (-1.80 - 1.04i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (10.8 - 6.25i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.98 + 1.14i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.21T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + (-5.15 - 8.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.81 - 5.09i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 8.57T + 89T^{2} \) |
| 97 | \( 1 + (3.78 + 6.56i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.02436450733878529557769440967, −9.158428070384393999380043464252, −8.179933390045502680689220606732, −7.62924606179578463277264950084, −6.27385768660992813662851793433, −5.84911459006678904418580279633, −4.98444492537123917972255530419, −4.04505429796918575837607237310, −2.47874273151619365781758202356, −1.31483162632247106154921173124,
0.58676946134562279387763331956, 1.92600442881147344274746793631, 3.82639545079980994967298633257, 4.51715203559823512619625614117, 5.05235632638176875064068059315, 6.57297277695287108486808830695, 6.98253702718242142139013988251, 7.68823204364584358006443239839, 9.025500126487744325497458108556, 9.762011520748112006844982047224