| L(s) = 1 | + (0.693 + 0.0606i)2-s + (−1.49 − 0.263i)4-s + (1.33 − 1.79i)5-s + (0.0498 + 0.186i)7-s + (−2.36 − 0.633i)8-s + (1.03 − 1.16i)10-s + (−1.69 − 2.93i)11-s + (−0.912 + 1.95i)13-s + (0.0232 + 0.132i)14-s + (1.24 + 0.454i)16-s + (0.604 − 6.91i)17-s + (−2.43 + 3.61i)19-s + (−2.47 + 2.31i)20-s + (−0.997 − 2.13i)22-s + (−4.28 + 2.99i)23-s + ⋯ |
| L(s) = 1 | + (0.490 + 0.0428i)2-s + (−0.746 − 0.131i)4-s + (0.599 − 0.800i)5-s + (0.0188 + 0.0703i)7-s + (−0.835 − 0.223i)8-s + (0.328 − 0.366i)10-s + (−0.511 − 0.885i)11-s + (−0.253 + 0.542i)13-s + (0.00622 + 0.0353i)14-s + (0.311 + 0.113i)16-s + (0.146 − 1.67i)17-s + (−0.559 + 0.828i)19-s + (−0.552 + 0.518i)20-s + (−0.212 − 0.455i)22-s + (−0.892 + 0.624i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.420587 - 0.957381i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.420587 - 0.957381i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.33 + 1.79i)T \) |
| 19 | \( 1 + (2.43 - 3.61i)T \) |
| good | 2 | \( 1 + (-0.693 - 0.0606i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-0.0498 - 0.186i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (1.69 + 2.93i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.912 - 1.95i)T + (-8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (-0.604 + 6.91i)T + (-16.7 - 2.95i)T^{2} \) |
| 23 | \( 1 + (4.28 - 2.99i)T + (7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (5.09 + 4.27i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.89 + 1.09i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.09 + 5.09i)T + 37iT^{2} \) |
| 41 | \( 1 + (-2.93 + 8.06i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.670 + 0.957i)T + (-14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (6.07 - 0.531i)T + (46.2 - 8.16i)T^{2} \) |
| 53 | \( 1 + (-6.81 - 9.72i)T + (-18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (-4.80 + 4.03i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.65 - 9.38i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (0.768 + 8.77i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (2.15 - 0.380i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (3.66 + 7.85i)T + (-46.9 + 55.9i)T^{2} \) |
| 79 | \( 1 + (-7.22 - 2.62i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (2.59 - 0.694i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (0.466 - 0.169i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-2.57 - 0.224i)T + (95.5 + 16.8i)T^{2} \) |
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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
−9.642456597855654747001994327541, −9.140642455630810718400221893934, −8.359867233514902822650144275074, −7.33067158754900401042307421610, −5.82509976633339571675553624737, −5.58713659489274024207565782264, −4.54801929841322327215373086852, −3.62548296527813932927339056064, −2.16495262584180378388477435947, −0.41951916003426682211018909147,
2.02448390636267079297510970640, 3.17161149517443311063208624440, 4.18850683310097786538082041746, 5.18248396025350584150651971531, 6.00973523453968679203165292079, 6.93696956081374518747337632844, 7.996778024644558910432332065830, 8.788711156137644420045146415854, 9.905376606542425045651203270294, 10.28944381635150169335645404606
