| L(s) = 1 | + (0.200 + 1.50i)3-s + (−0.247 − 0.109i)5-s + (1.16 − 2.49i)7-s + (0.677 − 0.183i)9-s + (2.49 − 3.20i)11-s + (0.271 − 0.529i)13-s + (0.114 − 0.393i)15-s + (−4.18 − 0.317i)17-s + (1.83 − 2.75i)19-s + (3.98 + 1.24i)21-s + (3.58 + 1.42i)23-s + (−3.31 − 3.64i)25-s + (2.17 + 5.17i)27-s + (7.40 − 2.94i)29-s + (−7.51 + 1.73i)31-s + ⋯ |
| L(s) = 1 | + (0.115 + 0.867i)3-s + (−0.110 − 0.0488i)5-s + (0.438 − 0.943i)7-s + (0.225 − 0.0612i)9-s + (0.753 − 0.965i)11-s + (0.0751 − 0.146i)13-s + (0.0296 − 0.101i)15-s + (−1.01 − 0.0770i)17-s + (0.420 − 0.632i)19-s + (0.869 + 0.271i)21-s + (0.747 + 0.297i)23-s + (−0.663 − 0.728i)25-s + (0.418 + 0.995i)27-s + (1.37 − 0.547i)29-s + (−1.34 + 0.311i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.69504 - 0.0774665i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69504 - 0.0774665i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + (8.23 - 9.95i)T \) |
| good | 3 | \( 1 + (-0.200 - 1.50i)T + (-2.89 + 0.785i)T^{2} \) |
| 5 | \( 1 + (0.247 + 0.109i)T + (3.36 + 3.69i)T^{2} \) |
| 7 | \( 1 + (-1.16 + 2.49i)T + (-4.51 - 5.35i)T^{2} \) |
| 11 | \( 1 + (-2.49 + 3.20i)T + (-2.67 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.271 + 0.529i)T + (-7.60 - 10.5i)T^{2} \) |
| 17 | \( 1 + (4.18 + 0.317i)T + (16.8 + 2.56i)T^{2} \) |
| 19 | \( 1 + (-1.83 + 2.75i)T + (-7.35 - 17.5i)T^{2} \) |
| 23 | \( 1 + (-3.58 - 1.42i)T + (16.7 + 15.7i)T^{2} \) |
| 29 | \( 1 + (-7.40 + 2.94i)T + (21.0 - 19.9i)T^{2} \) |
| 31 | \( 1 + (7.51 - 1.73i)T + (27.8 - 13.6i)T^{2} \) |
| 37 | \( 1 + (-5.34 - 1.45i)T + (31.9 + 18.7i)T^{2} \) |
| 41 | \( 1 + (-8.67 + 4.23i)T + (25.2 - 32.3i)T^{2} \) |
| 43 | \( 1 + (0.643 - 6.77i)T + (-42.2 - 8.08i)T^{2} \) |
| 47 | \( 1 + (-0.215 - 11.3i)T + (-46.9 + 1.77i)T^{2} \) |
| 53 | \( 1 + (0.737 + 4.28i)T + (-49.9 + 17.7i)T^{2} \) |
| 59 | \( 1 + (-1.52 + 0.116i)T + (58.3 - 8.89i)T^{2} \) |
| 61 | \( 1 + (7.21 - 7.35i)T + (-1.15 - 60.9i)T^{2} \) |
| 67 | \( 1 + (11.7 - 5.17i)T + (45.0 - 49.5i)T^{2} \) |
| 71 | \( 1 + (10.4 - 1.19i)T + (69.1 - 15.9i)T^{2} \) |
| 73 | \( 1 + (-11.5 + 8.66i)T + (20.4 - 70.0i)T^{2} \) |
| 79 | \( 1 + (-3.00 + 7.99i)T + (-59.4 - 52.0i)T^{2} \) |
| 83 | \( 1 + (-2.42 + 3.36i)T + (-26.2 - 78.7i)T^{2} \) |
| 89 | \( 1 + (-5.19 - 17.7i)T + (-75.0 + 47.8i)T^{2} \) |
| 97 | \( 1 + (-1.75 - 0.406i)T + (87.1 + 42.5i)T^{2} \) |
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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
−10.75362921282403306684417761638, −9.521449454252421841332443661262, −9.015819399836846137956449594255, −7.930729846099048757386602872051, −7.00501614263667460891487235873, −6.00382448841565181674785402041, −4.60541116516758264469018406086, −4.16564523192934927184903763807, −3.01335585916515942371566120748, −1.05765706732530444394891698004,
1.52712430938491032423507509496, 2.40030068478205636993383519941, 4.02028162827711788173924100801, 5.07033083766365247309296864264, 6.24011997034087856775744425377, 7.05884787819754490343019211502, 7.77922822768348168472642775647, 8.815644668152290656200751359385, 9.445593958472154741172912259236, 10.60816839661868931124668608278
