| L(s) = 1 | − 3-s − 4·7-s + 9-s − 3·13-s + 17-s + 8·19-s + 4·21-s − 8·23-s − 27-s − 9·29-s − 3·37-s + 3·39-s + 3·41-s − 8·43-s + 12·47-s + 9·49-s − 51-s − 11·53-s − 8·57-s − 2·61-s − 4·63-s + 4·67-s + 8·69-s + 6·73-s + 4·79-s + 81-s + 16·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.832·13-s + 0.242·17-s + 1.83·19-s + 0.872·21-s − 1.66·23-s − 0.192·27-s − 1.67·29-s − 0.493·37-s + 0.480·39-s + 0.468·41-s − 1.21·43-s + 1.75·47-s + 9/7·49-s − 0.140·51-s − 1.51·53-s − 1.05·57-s − 0.256·61-s − 0.503·63-s + 0.488·67-s + 0.963·69-s + 0.702·73-s + 0.450·79-s + 1/9·81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.60484515354383, −13.16615854356118, −12.43981246219630, −12.15322151438721, −11.99564229909519, −11.18511341018598, −10.78250721313711, −10.00845281869869, −9.892973164045775, −9.312253286440447, −9.157039655860561, −8.012093565796662, −7.746566243104701, −7.185334947714199, −6.711542542508649, −6.192412151279196, −5.618388107397012, −5.346613943197722, −4.659661036360419, −3.772412774217973, −3.613245358400256, −2.900149495953363, −2.239558373691145, −1.519701548854879, −0.5974461682649020, 0,
0.5974461682649020, 1.519701548854879, 2.239558373691145, 2.900149495953363, 3.613245358400256, 3.772412774217973, 4.659661036360419, 5.346613943197722, 5.618388107397012, 6.192412151279196, 6.711542542508649, 7.185334947714199, 7.746566243104701, 8.012093565796662, 9.157039655860561, 9.312253286440447, 9.892973164045775, 10.00845281869869, 10.78250721313711, 11.18511341018598, 11.99564229909519, 12.15322151438721, 12.43981246219630, 13.16615854356118, 13.60484515354383
