| L(s) = 1 | + 5-s + 7-s − 2·11-s − 4·13-s − 2·17-s − 2·19-s − 4·23-s + 25-s + 6·29-s − 2·31-s + 35-s − 10·37-s + 10·41-s − 12·43-s + 8·47-s + 49-s − 2·55-s − 8·59-s + 2·61-s − 4·65-s + 12·67-s + 10·71-s + 4·73-s − 2·77-s − 12·83-s − 2·85-s − 2·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.603·11-s − 1.10·13-s − 0.485·17-s − 0.458·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.359·31-s + 0.169·35-s − 1.64·37-s + 1.56·41-s − 1.82·43-s + 1.16·47-s + 1/7·49-s − 0.269·55-s − 1.04·59-s + 0.256·61-s − 0.496·65-s + 1.46·67-s + 1.18·71-s + 0.468·73-s − 0.227·77-s − 1.31·83-s − 0.216·85-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.598853455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.598853455\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.49457650564721, −15.27423034677904, −14.37635255304098, −14.10062619290619, −13.60053903676814, −12.79436298457035, −12.44329944716799, −11.90705129767718, −11.16123966465448, −10.62625521938105, −10.09508770139262, −9.639359832072511, −8.884086792194274, −8.327256818459410, −7.790535979669692, −7.045236849788275, −6.590077628058298, −5.758796060441471, −5.187275632659973, −4.649266664550718, −3.947641832464449, −2.978150113794086, −2.294047278284484, −1.752496479589447, −0.4978874787280279,
0.4978874787280279, 1.752496479589447, 2.294047278284484, 2.978150113794086, 3.947641832464449, 4.649266664550718, 5.187275632659973, 5.758796060441471, 6.590077628058298, 7.045236849788275, 7.790535979669692, 8.327256818459410, 8.884086792194274, 9.639359832072511, 10.09508770139262, 10.62625521938105, 11.16123966465448, 11.90705129767718, 12.44329944716799, 12.79436298457035, 13.60053903676814, 14.10062619290619, 14.37635255304098, 15.27423034677904, 15.49457650564721
