| L(s) = 1 | + (1.37 − 0.328i)2-s + (0.516 + 3.26i)3-s + (1.78 − 0.903i)4-s + (−0.0328 − 2.23i)5-s + (1.78 + 4.31i)6-s + 1.19i·7-s + (2.15 − 1.82i)8-s + (−7.52 + 2.44i)9-s + (−0.779 − 3.06i)10-s + (2.63 + 5.17i)11-s + (3.86 + 5.35i)12-s + (1.38 − 2.71i)13-s + (0.392 + 1.64i)14-s + (7.27 − 1.26i)15-s + (2.36 − 3.22i)16-s + (1.03 − 0.751i)17-s + ⋯ |
| L(s) = 1 | + (0.972 − 0.232i)2-s + (0.298 + 1.88i)3-s + (0.892 − 0.451i)4-s + (−0.0146 − 0.999i)5-s + (0.727 + 1.76i)6-s + 0.452i·7-s + (0.762 − 0.646i)8-s + (−2.50 + 0.814i)9-s + (−0.246 − 0.969i)10-s + (0.794 + 1.55i)11-s + (1.11 + 1.54i)12-s + (0.384 − 0.753i)13-s + (0.105 + 0.440i)14-s + (1.87 − 0.325i)15-s + (0.592 − 0.805i)16-s + (0.250 − 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.31982 + 1.24078i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31982 + 1.24078i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.37 + 0.328i)T \) |
| 5 | \( 1 + (0.0328 + 2.23i)T \) |
| good | 3 | \( 1 + (-0.516 - 3.26i)T + (-2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 - 1.19iT - 7T^{2} \) |
| 11 | \( 1 + (-2.63 - 5.17i)T + (-6.46 + 8.89i)T^{2} \) |
| 13 | \( 1 + (-1.38 + 2.71i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.03 + 0.751i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (4.56 + 0.722i)T + (18.0 + 5.87i)T^{2} \) |
| 23 | \( 1 + (0.833 + 0.270i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (0.988 + 6.23i)T + (-27.5 + 8.96i)T^{2} \) |
| 31 | \( 1 + (0.991 - 0.720i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.69 - 0.865i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-4.73 + 1.53i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-2.77 + 2.77i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.49 + 3.26i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (11.5 - 1.83i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (0.622 + 0.317i)T + (34.6 + 47.7i)T^{2} \) |
| 61 | \( 1 + (-9.63 + 4.90i)T + (35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (11.8 + 1.87i)T + (63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (2.89 - 3.97i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.69 - 1.19i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.62 - 4.08i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (5.60 + 0.887i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-3.34 - 1.08i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-12.1 - 8.81i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.45805575007863222073222668927, −10.44176438063594775385360543064, −9.712541286046424053787506030427, −9.034189727690917164891289406127, −7.903462798003222873395453089599, −6.12280538714234692383721458054, −5.17584442642708030263216761532, −4.43965247630831904670505217878, −3.79081917551339962800652291282, −2.30356913678298348188803264394,
1.58075023472794664221721550038, 2.89131388578550165538232775787, 3.80148591383594326622365669612, 5.99453695901594034809135000386, 6.33123343836675590285948044791, 7.13034612666661140512887646378, 7.955110885004542221156476799281, 8.848310728197836657316569343419, 10.90077528582358452223081426443, 11.32163416068917007702756290837
