| L(s) = 1 | + (0.528 − 0.0251i)2-s + (−0.296 + 1.70i)3-s + (−1.71 + 0.163i)4-s + (0.902 + 0.860i)5-s + (−0.113 + 0.909i)6-s + (0.405 + 2.10i)7-s + (−1.94 + 0.280i)8-s + (−2.82 − 1.01i)9-s + (0.498 + 0.432i)10-s + (−0.599 + 0.308i)11-s + (0.229 − 2.97i)12-s + (−3.62 − 0.698i)13-s + (0.267 + 1.10i)14-s + (−1.73 + 1.28i)15-s + (2.35 − 0.453i)16-s + (3.17 + 6.96i)17-s + ⋯ |
| L(s) = 1 | + (0.373 − 0.0178i)2-s + (−0.171 + 0.985i)3-s + (−0.856 + 0.0817i)4-s + (0.403 + 0.384i)5-s + (−0.0465 + 0.371i)6-s + (0.153 + 0.794i)7-s + (−0.688 + 0.0990i)8-s + (−0.941 − 0.337i)9-s + (0.157 + 0.136i)10-s + (−0.180 + 0.0931i)11-s + (0.0662 − 0.857i)12-s + (−1.00 − 0.193i)13-s + (0.0714 + 0.294i)14-s + (−0.448 + 0.331i)15-s + (0.588 − 0.113i)16-s + (0.771 + 1.68i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.583710 + 0.885873i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.583710 + 0.885873i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.296 - 1.70i)T \) |
| 23 | \( 1 + (1.36 + 4.59i)T \) |
| good | 2 | \( 1 + (-0.528 + 0.0251i)T + (1.99 - 0.190i)T^{2} \) |
| 5 | \( 1 + (-0.902 - 0.860i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.405 - 2.10i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (0.599 - 0.308i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (3.62 + 0.698i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-3.17 - 6.96i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-7.63 - 3.48i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.453 - 4.74i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (-2.97 + 2.34i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (-2.32 + 7.91i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (6.70 - 7.03i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-4.55 + 5.78i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (-4.87 - 2.81i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.25 - 1.45i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (0.333 - 1.73i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (3.84 - 9.59i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (0.421 - 0.817i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (3.21 + 5.00i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.04 + 2.29i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (2.71 - 0.940i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (1.49 - 1.42i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (0.539 - 3.75i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-1.29 - 0.314i)T + (86.2 + 44.4i)T^{2} \) |
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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
−12.44194512649158408551497781089, −12.03437010223911547861960988499, −10.44399693821937955182469870119, −9.906063813231947201147092290978, −8.926561829250972396078175033098, −7.928302441979457349879522713671, −5.92585264481488651163239140945, −5.36512498253541551177838322929, −4.12805640543349295980706864144, −2.87038942477033989248449546463,
0.901258777843582052160852181826, 3.05736995223488395872679257287, 4.90724580260656576441179099969, 5.50465167410723533517449086790, 7.11699264237310752706834050016, 7.80325219517648185090121540277, 9.263389375799849835850766926742, 9.882598941300035702106272385270, 11.56893516575935977980811884034, 12.13233937044828583141010584506
