| L(s) = 1 | + 2·3-s − 4·5-s − 7-s + 9-s + 11-s − 2·13-s − 8·15-s − 2·17-s − 2·21-s + 8·23-s + 11·25-s − 4·27-s − 29-s + 2·33-s + 4·35-s − 6·37-s − 4·39-s − 6·41-s + 4·43-s − 4·45-s + 12·47-s + 49-s − 4·51-s − 2·53-s − 4·55-s + 8·59-s − 4·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 2.06·15-s − 0.485·17-s − 0.436·21-s + 1.66·23-s + 11/5·25-s − 0.769·27-s − 0.185·29-s + 0.348·33-s + 0.676·35-s − 0.986·37-s − 0.640·39-s − 0.937·41-s + 0.609·43-s − 0.596·45-s + 1.75·47-s + 1/7·49-s − 0.560·51-s − 0.274·53-s − 0.539·55-s + 1.04·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
| 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
−15.11803056535642, −14.75729193104832, −14.34114330919132, −13.58794137856695, −13.15832418087671, −12.53585794465466, −12.03764689931738, −11.60383869401136, −10.94832203205507, −10.55207298686373, −9.668541189077722, −9.101221564878513, −8.643192638311294, −8.361721031484831, −7.537604740706544, −7.188255407244259, −6.862853053799928, −5.818919729094812, −5.004594758393946, −4.363012410267492, −3.860387725440072, −3.198560179940740, −2.896565034968945, −2.013060955483343, −0.8858227806652953, 0,
0.8858227806652953, 2.013060955483343, 2.896565034968945, 3.198560179940740, 3.860387725440072, 4.363012410267492, 5.004594758393946, 5.818919729094812, 6.862853053799928, 7.188255407244259, 7.537604740706544, 8.361721031484831, 8.643192638311294, 9.101221564878513, 9.668541189077722, 10.55207298686373, 10.94832203205507, 11.60383869401136, 12.03764689931738, 12.53585794465466, 13.15832418087671, 13.58794137856695, 14.34114330919132, 14.75729193104832, 15.11803056535642
