| L(s) = 1 | + (−0.464 − 1.33i)2-s + (−0.540 − 0.841i)3-s + (−1.56 + 1.24i)4-s + (0.798 − 0.364i)5-s + (−0.872 + 1.11i)6-s + (2.23 − 0.657i)7-s + (2.38 + 1.51i)8-s + (−0.415 + 0.909i)9-s + (−0.858 − 0.897i)10-s + (−0.502 + 0.435i)11-s + (1.89 + 0.648i)12-s + (0.537 − 1.82i)13-s + (−1.91 − 2.68i)14-s + (−0.738 − 0.474i)15-s + (0.921 − 3.89i)16-s + (0.515 − 3.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.328 − 0.944i)2-s + (−0.312 − 0.485i)3-s + (−0.784 + 0.620i)4-s + (0.357 − 0.163i)5-s + (−0.356 + 0.454i)6-s + (0.846 − 0.248i)7-s + (0.843 + 0.537i)8-s + (−0.138 + 0.303i)9-s + (−0.271 − 0.283i)10-s + (−0.151 + 0.131i)11-s + (0.546 + 0.187i)12-s + (0.148 − 0.507i)13-s + (−0.512 − 0.717i)14-s + (−0.190 − 0.122i)15-s + (0.230 − 0.973i)16-s + (0.125 − 0.870i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.515663 - 1.03291i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.515663 - 1.03291i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.464 + 1.33i)T \) |
| 3 | \( 1 + (0.540 + 0.841i)T \) |
| 23 | \( 1 + (-2.85 + 3.85i)T \) |
| good | 5 | \( 1 + (-0.798 + 0.364i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-2.23 + 0.657i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (0.502 - 0.435i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.537 + 1.82i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.515 + 3.58i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-7.17 + 1.03i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (7.91 + 1.13i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.262 - 0.168i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-5.02 - 2.29i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (4.68 + 10.2i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (4.26 + 6.63i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 1.07T + 47T^{2} \) |
| 53 | \( 1 + (-1.94 - 6.62i)T + (-44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-1.87 + 6.38i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (3.59 - 5.59i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-11.3 - 9.87i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-0.962 + 1.11i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.90 + 13.2i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-7.03 - 2.06i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-2.45 - 1.12i)T + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-6.06 + 3.90i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (2.94 + 6.43i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.61295301795618167050241683691, −9.688025598886332834011771692965, −8.877188435786127419520297121217, −7.77993797544189000364803114273, −7.23997667574482402617244773156, −5.50665395188123836354467363444, −4.89078500812483399932427454756, −3.42405204521231561653408510467, −2.10963517954203843504394074285, −0.880626669844135234137564240820,
1.54183095473503405347657473409, 3.64278675288513500556559417703, 4.89330126871645278296678651889, 5.57586839554267352828198331642, 6.45143708825701624499449778575, 7.62793235255399924031053262935, 8.312486766898108076025530483598, 9.452361963717743978221792962973, 9.872057536031725870134784144964, 11.05986754576494085499552954584
