L(s) = 1 | − 7-s + 4·11-s + 3·13-s + 7·17-s + 6·19-s + 9·23-s + 3·29-s − 7·31-s + 10·37-s + 41-s − 13·43-s − 2·47-s + 49-s + 53-s − 11·59-s − 13·61-s − 8·71-s + 8·73-s − 4·77-s + 4·79-s − 7·83-s + 14·89-s − 3·91-s + 8·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 1.20·11-s + 0.832·13-s + 1.69·17-s + 1.37·19-s + 1.87·23-s + 0.557·29-s − 1.25·31-s + 1.64·37-s + 0.156·41-s − 1.98·43-s − 0.291·47-s + 1/7·49-s + 0.137·53-s − 1.43·59-s − 1.66·61-s − 0.949·71-s + 0.936·73-s − 0.455·77-s + 0.450·79-s − 0.768·83-s + 1.48·89-s − 0.314·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.901418712\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.901418712\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.66643732339124, −13.36643869930741, −12.74141845358576, −12.23763335637450, −11.83090111479301, −11.30011086342004, −10.92484846919689, −10.25352642152042, −9.652826457795699, −9.360517463663003, −8.904721368360482, −8.299182881437399, −7.556369164939299, −7.360045935677354, −6.570157780876113, −6.203206714419819, −5.601229212844268, −5.062652111535557, −4.475628514577794, −3.658228730906535, −3.193264705778436, −3.012263381915309, −1.702807736411274, −1.223420391174653, −0.7138016707083573,
0.7138016707083573, 1.223420391174653, 1.702807736411274, 3.012263381915309, 3.193264705778436, 3.658228730906535, 4.475628514577794, 5.062652111535557, 5.601229212844268, 6.203206714419819, 6.570157780876113, 7.360045935677354, 7.556369164939299, 8.299182881437399, 8.904721368360482, 9.360517463663003, 9.652826457795699, 10.25352642152042, 10.92484846919689, 11.30011086342004, 11.83090111479301, 12.23763335637450, 12.74141845358576, 13.36643869930741, 13.66643732339124