L(s) = 1 | + (1.69 − 2.92i)3-s + (−3.56 + 0.954i)5-s + (−1.92 − 1.11i)7-s + (−4.21 − 7.30i)9-s + 2.24·11-s + (−1.64 + 0.440i)13-s + (−3.22 + 12.0i)15-s + (−1.39 + 5.22i)17-s + (−0.234 − 0.873i)19-s + (−6.51 + 3.76i)21-s + (−2.13 + 2.13i)23-s + (7.44 − 4.30i)25-s − 18.3·27-s + (4.32 − 4.32i)29-s + (−2.78 − 2.78i)31-s + ⋯ |
L(s) = 1 | + (0.976 − 1.69i)3-s + (−1.59 + 0.426i)5-s + (−0.728 − 0.420i)7-s + (−1.40 − 2.43i)9-s + 0.676·11-s + (−0.455 + 0.122i)13-s + (−0.833 + 3.11i)15-s + (−0.339 + 1.26i)17-s + (−0.0537 − 0.200i)19-s + (−1.42 + 0.821i)21-s + (−0.446 + 0.446i)23-s + (1.48 − 0.860i)25-s − 3.53·27-s + (0.802 − 0.802i)29-s + (−0.499 − 0.499i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0953673 + 0.763781i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0953673 + 0.763781i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 + (0.473 + 6.06i)T \) |
good | 3 | \( 1 + (-1.69 + 2.92i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (3.56 - 0.954i)T + (4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (1.92 + 1.11i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + (1.64 - 0.440i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (1.39 - 5.22i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.234 + 0.873i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (2.13 - 2.13i)T - 23iT^{2} \) |
| 29 | \( 1 + (-4.32 + 4.32i)T - 29iT^{2} \) |
| 31 | \( 1 + (2.78 + 2.78i)T + 31iT^{2} \) |
| 41 | \( 1 + (10.1 + 5.86i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.347 - 0.347i)T - 43iT^{2} \) |
| 47 | \( 1 + 9.31iT - 47T^{2} \) |
| 53 | \( 1 + (-1.97 - 3.42i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.10 - 1.90i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-0.0255 - 0.0954i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (1.46 - 2.54i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.21 - 4.74i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 6.21iT - 73T^{2} \) |
| 79 | \( 1 + (4.01 + 14.9i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-7.19 + 4.15i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (8.39 + 2.24i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (6.02 + 6.02i)T + 97iT^{2} \) |
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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
−10.18669944473129253021663125612, −8.901525667939911767130662154401, −8.290139675837327711885703840782, −7.42902743837068419271939396833, −6.95681693063541686549403964736, −6.17610220378728615637429786628, −3.92273392516822490606070115561, −3.43983499856139242861375931788, −2.08126556237328799322353829318, −0.36870104524461732113546350264,
2.86093519305067826817041488323, 3.54073311301052490400474921012, 4.48218089628954355349740653548, 5.09970711304292440643253216287, 6.85454470884779273991526604680, 8.056051664974204249404853007718, 8.612037285727675307460858939604, 9.382829408998679141939200024075, 10.03872515354083122752226039154, 11.11594746835594485697628111366