L(s) = 1 | + (0.866 + 0.5i)2-s + (−1.60 + 0.923i)3-s + (0.499 + 0.866i)4-s + (2.80 + 1.61i)5-s − 1.84·6-s + 0.999i·8-s + (0.207 − 0.358i)9-s + (1.61 + 2.80i)10-s + (−2.04 − 2.60i)11-s + (−1.60 − 0.923i)12-s − 6.10·13-s − 5.98·15-s + (−0.5 + 0.866i)16-s + (4.06 + 7.04i)17-s + (0.358 − 0.207i)18-s + (−3.30 + 5.72i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.923 + 0.533i)3-s + (0.249 + 0.433i)4-s + (1.25 + 0.723i)5-s − 0.754·6-s + 0.353i·8-s + (0.0690 − 0.119i)9-s + (0.511 + 0.886i)10-s + (−0.617 − 0.786i)11-s + (−0.461 − 0.266i)12-s − 1.69·13-s − 1.54·15-s + (−0.125 + 0.216i)16-s + (0.986 + 1.70i)17-s + (0.0845 − 0.0488i)18-s + (−0.758 + 1.31i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.455852997\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.455852997\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (2.04 + 2.60i)T \) |
good | 3 | \( 1 + (1.60 - 0.923i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.80 - 1.61i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 6.10T + 13T^{2} \) |
| 17 | \( 1 + (-4.06 - 7.04i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.30 - 5.72i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.65 + 4.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.32iT - 29T^{2} \) |
| 31 | \( 1 + (2.03 - 1.17i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.64 - 4.57i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 1.22T + 41T^{2} \) |
| 43 | \( 1 - 3.73iT - 43T^{2} \) |
| 47 | \( 1 + (-1.47 - 0.853i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.99 - 3.44i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.82 + 4.51i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.27 - 7.41i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.17 + 2.03i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.17T + 71T^{2} \) |
| 73 | \( 1 + (5.49 + 9.51i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.34 - 4.24i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 5.60T + 83T^{2} \) |
| 89 | \( 1 + (4.03 + 2.33i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.82iT - 97T^{2} \) |
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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
−10.42301804785480221193841132169, −9.881536542297813069834139041302, −8.488088287110751577359888664391, −7.62958263640464855335811803663, −6.44415667571240172373903714620, −5.89900684004623339968489137661, −5.37546065697967680699659518593, −4.40821097425697790774503227423, −3.09615317742517290043210486109, −2.06926837897335537490159317969,
0.54936287064453547936332008938, 1.94976458439324941192798047197, 2.83131272050060018546159905118, 4.73867013381239741191753383662, 5.22546196643521189800377071459, 5.69392557089679622658447787493, 7.06382229931023782457140468714, 7.25876095319542898696808251859, 9.077564431411005530108348594979, 9.613240633464637136723037392301