| L(s) = 1 | − 11-s + 2·13-s + 2·17-s + 4·19-s − 23-s + 8·29-s + 4·31-s − 2·37-s − 4·41-s − 8·43-s + 8·47-s − 7·49-s − 4·53-s + 4·59-s + 2·61-s + 10·67-s − 6·73-s − 4·79-s − 4·83-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.208·23-s + 1.48·29-s + 0.718·31-s − 0.328·37-s − 0.624·41-s − 1.21·43-s + 1.16·47-s − 49-s − 0.549·53-s + 0.520·59-s + 0.256·61-s + 1.22·67-s − 0.702·73-s − 0.450·79-s − 0.439·83-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.23220440921632, −12.69964230338843, −12.17857603116560, −11.77023151195700, −11.44085249032099, −10.80956833608235, −10.31215988357282, −9.915488039585634, −9.616286111924745, −8.761679499351870, −8.539684893741044, −7.987549281034981, −7.587976859795563, −6.861558023852100, −6.606450642801047, −5.912897400602353, −5.519915134887117, −4.837199443471238, −4.595477920838626, −3.681340576880334, −3.364960273552215, −2.770628806244535, −2.159125398689524, −1.335989207143141, −0.9261843605654017, 0,
0.9261843605654017, 1.335989207143141, 2.159125398689524, 2.770628806244535, 3.364960273552215, 3.681340576880334, 4.595477920838626, 4.837199443471238, 5.519915134887117, 5.912897400602353, 6.606450642801047, 6.861558023852100, 7.587976859795563, 7.987549281034981, 8.539684893741044, 8.761679499351870, 9.616286111924745, 9.915488039585634, 10.31215988357282, 10.80956833608235, 11.44085249032099, 11.77023151195700, 12.17857603116560, 12.69964230338843, 13.23220440921632
