| L(s) = 1 | − 3-s − 2·7-s + 9-s − 11-s − 17-s + 2·21-s − 4·23-s − 27-s + 2·29-s − 4·31-s + 33-s + 2·37-s − 6·43-s − 3·49-s + 51-s + 6·53-s − 14·59-s − 2·61-s − 2·63-s + 14·67-s + 4·69-s + 2·71-s − 16·73-s + 2·77-s − 12·79-s + 81-s − 2·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.242·17-s + 0.436·21-s − 0.834·23-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.174·33-s + 0.328·37-s − 0.914·43-s − 3/7·49-s + 0.140·51-s + 0.824·53-s − 1.82·59-s − 0.256·61-s − 0.251·63-s + 1.71·67-s + 0.481·69-s + 0.237·71-s − 1.87·73-s + 0.227·77-s − 1.35·79-s + 1/9·81-s − 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
| 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
−13.43386960174461, −12.79086082350083, −12.57897117974052, −12.03700501859184, −11.57330362299973, −11.08898783657844, −10.66535455998813, −10.07823244333675, −9.818880888651264, −9.321264115340519, −8.716728314996461, −8.233937962315143, −7.717202707183318, −7.109757782547025, −6.755120737095606, −6.189442712471375, −5.788154326549254, −5.299554719016453, −4.678607484801715, −4.170543124979615, −3.638566421244777, −2.998170185287489, −2.491382664420458, −1.716902000143306, −1.144698069861360, 0, 0,
1.144698069861360, 1.716902000143306, 2.491382664420458, 2.998170185287489, 3.638566421244777, 4.170543124979615, 4.678607484801715, 5.299554719016453, 5.788154326549254, 6.189442712471375, 6.755120737095606, 7.109757782547025, 7.717202707183318, 8.233937962315143, 8.716728314996461, 9.321264115340519, 9.818880888651264, 10.07823244333675, 10.66535455998813, 11.08898783657844, 11.57330362299973, 12.03700501859184, 12.57897117974052, 12.79086082350083, 13.43386960174461
