L(s) = 1 | + (−0.707 + 0.707i)2-s − i·3-s − 1.00i·4-s + (−2.16 − 2.16i)5-s + (0.707 + 0.707i)6-s + (2.62 − 0.292i)7-s + (0.707 + 0.707i)8-s − 9-s + 3.06·10-s + (0.516 + 0.516i)11-s − 1.00·12-s + (3.60 − 0.0469i)13-s + (−1.65 + 2.06i)14-s + (−2.16 + 2.16i)15-s − 1.00·16-s − 2.57·17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.577i·3-s − 0.500i·4-s + (−0.969 − 0.969i)5-s + (0.288 + 0.288i)6-s + (0.993 − 0.110i)7-s + (0.250 + 0.250i)8-s − 0.333·9-s + 0.969·10-s + (0.155 + 0.155i)11-s − 0.288·12-s + (0.999 − 0.0130i)13-s + (−0.441 + 0.552i)14-s + (−0.559 + 0.559i)15-s − 0.250·16-s − 0.624·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.448769 - 0.653605i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.448769 - 0.653605i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (-2.62 + 0.292i)T \) |
| 13 | \( 1 + (-3.60 + 0.0469i)T \) |
good | 5 | \( 1 + (2.16 + 2.16i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.516 - 0.516i)T + 11iT^{2} \) |
| 17 | \( 1 + 2.57T + 17T^{2} \) |
| 19 | \( 1 + (3.82 + 3.82i)T + 19iT^{2} \) |
| 23 | \( 1 + 6.97iT - 23T^{2} \) |
| 29 | \( 1 + 6.32T + 29T^{2} \) |
| 31 | \( 1 + (4.33 + 4.33i)T + 31iT^{2} \) |
| 37 | \( 1 + (3.58 + 3.58i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.176 + 0.176i)T + 41iT^{2} \) |
| 43 | \( 1 - 7.89iT - 43T^{2} \) |
| 47 | \( 1 + (-1.41 + 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 + 0.514T + 53T^{2} \) |
| 59 | \( 1 + (3.17 - 3.17i)T - 59iT^{2} \) |
| 61 | \( 1 - 6.41iT - 61T^{2} \) |
| 67 | \( 1 + (3.96 - 3.96i)T - 67iT^{2} \) |
| 71 | \( 1 + (-8.12 + 8.12i)T - 71iT^{2} \) |
| 73 | \( 1 + (-11.6 + 11.6i)T - 73iT^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + (-1.01 - 1.01i)T + 83iT^{2} \) |
| 89 | \( 1 + (-1.10 + 1.10i)T - 89iT^{2} \) |
| 97 | \( 1 + (9.34 + 9.34i)T + 97iT^{2} \) |
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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
−10.87140111184602596684161187555, −9.143600084395631030553712742806, −8.595610067709012154257147863807, −7.985271452779885404090175570655, −7.13363742136715842307920219558, −6.08577856744845749437963509708, −4.85452600993166717075466883423, −4.08307034644370563219510960333, −1.95472897536364406063810010244, −0.55118220855173462400731389571,
1.85674640290608875164837534144, 3.48552596847837310251339022203, 3.96732833121641968829494732540, 5.40457148306889750099931758824, 6.74594565758980879084041606589, 7.76329497485876543916616050497, 8.434056772785093145979829570609, 9.301585264589211097152104418336, 10.54480079899871866841962415805, 11.04490061116561557794026618902