| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − i·8-s − 9-s + 4·11-s + i·12-s − 2i·13-s + 16-s + i·17-s − i·18-s + 4·19-s + 4i·22-s − 4i·23-s − 24-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + 1.20·11-s + 0.288i·12-s − 0.554i·13-s + 0.250·16-s + 0.242i·17-s − 0.235i·18-s + 0.917·19-s + 0.852i·22-s − 0.834i·23-s − 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.682861287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.682861287\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 - iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.674976872205014579610399513176, −8.035609656333998927155692578483, −7.17973435817003348097957125423, −6.67711219316901086933526183224, −5.84623157877719780983481316184, −5.15512757928069106842408712785, −4.05414912395753202368217812084, −3.23934968937428611353574434117, −1.86874871860044182921554026911, −0.66659293232709081740859685894,
1.11452878732348489724099533534, 2.20686752998281625449359413655, 3.49275402817289348986607960888, 3.86206364322777521341628953009, 4.93038821072975749988228583327, 5.59984040164011358027007334065, 6.68157484600785741437958656523, 7.42637657357619837608742266578, 8.561070558430045159539506325686, 9.102686175362718376924115419268
