| L(s) = 1 | − 3·7-s + 5·11-s + 13-s + 5·17-s − 4·19-s + 2·23-s + 9·29-s − 3·31-s − 10·37-s + 12·41-s − 2·43-s + 9·47-s + 2·49-s − 9·53-s + 3·59-s − 7·61-s − 9·67-s − 10·73-s − 15·77-s + 10·79-s + 83-s + 4·89-s − 3·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 1.50·11-s + 0.277·13-s + 1.21·17-s − 0.917·19-s + 0.417·23-s + 1.67·29-s − 0.538·31-s − 1.64·37-s + 1.87·41-s − 0.304·43-s + 1.31·47-s + 2/7·49-s − 1.23·53-s + 0.390·59-s − 0.896·61-s − 1.09·67-s − 1.17·73-s − 1.70·77-s + 1.12·79-s + 0.109·83-s + 0.423·89-s − 0.314·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.15330908386782, −13.57889929501092, −13.07942465533651, −12.46710089805388, −12.09983544246154, −11.93204840216545, −10.91669925941156, −10.68975197363345, −10.08682450063055, −9.541421656775066, −9.092840061080465, −8.764838573910365, −8.081881729197846, −7.427342705601145, −6.883080934020277, −6.396874790046080, −6.076561939665208, −5.462546568970468, −4.641406315138965, −4.122297912180504, −3.518814369256419, −3.120491719821569, −2.404603185202803, −1.455499067654707, −0.9769842621047078, 0,
0.9769842621047078, 1.455499067654707, 2.404603185202803, 3.120491719821569, 3.518814369256419, 4.122297912180504, 4.641406315138965, 5.462546568970468, 6.076561939665208, 6.396874790046080, 6.883080934020277, 7.427342705601145, 8.081881729197846, 8.764838573910365, 9.092840061080465, 9.541421656775066, 10.08682450063055, 10.68975197363345, 10.91669925941156, 11.93204840216545, 12.09983544246154, 12.46710089805388, 13.07942465533651, 13.57889929501092, 14.15330908386782
