| L(s) = 1 | + 7-s + 11-s + 2·13-s − 4·17-s + 2·19-s + 5·23-s − 29-s + 2·31-s + 3·37-s − 12·41-s − 11·43-s + 2·47-s + 49-s − 6·53-s − 10·59-s + 4·61-s − 67-s − 3·71-s + 77-s + 9·79-s − 2·83-s + 6·89-s + 2·91-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.301·11-s + 0.554·13-s − 0.970·17-s + 0.458·19-s + 1.04·23-s − 0.185·29-s + 0.359·31-s + 0.493·37-s − 1.87·41-s − 1.67·43-s + 0.291·47-s + 1/7·49-s − 0.824·53-s − 1.30·59-s + 0.512·61-s − 0.122·67-s − 0.356·71-s + 0.113·77-s + 1.01·79-s − 0.219·83-s + 0.635·89-s + 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.45505517098127, −15.17874395416315, −14.64565118948190, −13.91152260550703, −13.47639785035392, −13.11057888206615, −12.36317103135977, −11.71614107115731, −11.37895708563896, −10.78129143512737, −10.26159380781277, −9.521368383441794, −9.013170823208882, −8.480917124326303, −7.942644286569231, −7.219996963492592, −6.579925209377557, −6.231610779154049, −5.174785684573715, −4.930669402599107, −4.085582722453782, −3.414450045319946, −2.739764421212071, −1.787959928658741, −1.180516456498978, 0,
1.180516456498978, 1.787959928658741, 2.739764421212071, 3.414450045319946, 4.085582722453782, 4.930669402599107, 5.174785684573715, 6.231610779154049, 6.579925209377557, 7.219996963492592, 7.942644286569231, 8.480917124326303, 9.013170823208882, 9.521368383441794, 10.26159380781277, 10.78129143512737, 11.37895708563896, 11.71614107115731, 12.36317103135977, 13.11057888206615, 13.47639785035392, 13.91152260550703, 14.64565118948190, 15.17874395416315, 15.45505517098127
