| L(s) = 1 | + (0.240 + 0.277i)2-s + (−0.500 − 0.321i)3-s + (0.265 − 1.84i)4-s + (−1.01 + 2.21i)5-s + (−0.0310 − 0.215i)6-s + (0.240 + 0.0707i)7-s + (1.19 − 0.766i)8-s + (−1.09 − 2.40i)9-s + (−0.858 + 0.251i)10-s + (−0.00585 + 0.00676i)11-s + (−0.726 + 0.838i)12-s + (1.01 − 0.297i)13-s + (0.0382 + 0.0837i)14-s + (1.21 − 0.783i)15-s + (−3.08 − 0.904i)16-s + (−0.511 − 3.55i)17-s + ⋯ |
| L(s) = 1 | + (0.169 + 0.196i)2-s + (−0.288 − 0.185i)3-s + (0.132 − 0.923i)4-s + (−0.452 + 0.991i)5-s + (−0.0126 − 0.0881i)6-s + (0.0910 + 0.0267i)7-s + (0.421 − 0.271i)8-s + (−0.366 − 0.802i)9-s + (−0.271 + 0.0796i)10-s + (−0.00176 + 0.00203i)11-s + (−0.209 + 0.241i)12-s + (0.280 − 0.0824i)13-s + (0.0102 + 0.0223i)14-s + (0.314 − 0.202i)15-s + (−0.770 − 0.226i)16-s + (−0.123 − 0.862i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0526 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0526 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.770457 - 0.812126i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.770457 - 0.812126i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (0.375 + 4.78i)T \) |
| 29 | \( 1 + (-0.142 - 0.989i)T \) |
| good | 2 | \( 1 + (-0.240 - 0.277i)T + (-0.284 + 1.97i)T^{2} \) |
| 3 | \( 1 + (0.500 + 0.321i)T + (1.24 + 2.72i)T^{2} \) |
| 5 | \( 1 + (1.01 - 2.21i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.240 - 0.0707i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (0.00585 - 0.00676i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.01 + 0.297i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.511 + 3.55i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.529 + 3.68i)T + (-18.2 - 5.35i)T^{2} \) |
| 31 | \( 1 + (-0.633 + 0.407i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (4.00 + 8.76i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.42 + 3.11i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (4.05 + 2.60i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 + (-8.62 - 2.53i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (2.66 - 0.782i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-7.85 + 5.05i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (8.84 + 10.2i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-9.96 - 11.5i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.75 - 12.1i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (3.41 - 1.00i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.780 + 1.70i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-12.2 - 7.87i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (4.96 - 10.8i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.49875585805071663725105433430, −9.498685979904278709377510401299, −8.643674418648591944108362941582, −7.15841051561659197599097226396, −6.83660593177091917218768559373, −5.89591016685463099930037388547, −4.95568864124044409818775607366, −3.65585723613336005944947924920, −2.43779577992197790315674819548, −0.58934920034658858888117288950,
1.72483749248047598517538628766, 3.27307868779499724170732195913, 4.28118572889960103546565887344, 5.03506405216783997189243929626, 6.15562756257674622297138601710, 7.51037127293693742184987051140, 8.233220702840862931755740532843, 8.682866088764389672437719724709, 9.989195471124021510255569465874, 10.92633113965543423066526399648
