| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s − 4·11-s + 12-s − 2·13-s − 14-s + 15-s + 16-s + 2·17-s + 18-s + 4·19-s + 20-s − 21-s − 4·22-s + 8·23-s + 24-s + 25-s − 2·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.05425800031988, −12.70926212515438, −12.29951213020296, −11.69930093398636, −11.28189265111711, −10.57278579299644, −10.24435643304949, −9.991367352038099, −9.222417532742731, −8.940177138582902, −8.400691140386946, −7.626564948869277, −7.359742753428512, −7.062805432931462, −6.367951683818868, −5.681205562213961, −5.424778302948164, −4.827057628704633, −4.550768252778331, −3.531884151779686, −3.228233003461085, −2.862437339976343, −2.242561829789037, −1.653645945744498, −0.9464211583203394, 0,
0.9464211583203394, 1.653645945744498, 2.242561829789037, 2.862437339976343, 3.228233003461085, 3.531884151779686, 4.550768252778331, 4.827057628704633, 5.424778302948164, 5.681205562213961, 6.367951683818868, 7.062805432931462, 7.359742753428512, 7.626564948869277, 8.400691140386946, 8.940177138582902, 9.222417532742731, 9.991367352038099, 10.24435643304949, 10.57278579299644, 11.28189265111711, 11.69930093398636, 12.29951213020296, 12.70926212515438, 13.05425800031988
