| L(s) = 1 | − 7-s − 4·11-s + 3·13-s − 7·17-s + 6·19-s − 9·23-s − 3·29-s − 7·31-s + 10·37-s − 41-s − 13·43-s + 2·47-s + 49-s − 53-s + 11·59-s − 13·61-s + 8·71-s + 8·73-s + 4·77-s + 4·79-s + 7·83-s − 14·89-s − 3·91-s + 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s + 0.832·13-s − 1.69·17-s + 1.37·19-s − 1.87·23-s − 0.557·29-s − 1.25·31-s + 1.64·37-s − 0.156·41-s − 1.98·43-s + 0.291·47-s + 1/7·49-s − 0.137·53-s + 1.43·59-s − 1.66·61-s + 0.949·71-s + 0.936·73-s + 0.455·77-s + 0.450·79-s + 0.768·83-s − 1.48·89-s − 0.314·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5389290551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5389290551\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.58461690049298, −13.31612034515179, −12.92694288683306, −12.34733760645245, −11.63221975411945, −11.37746999212027, −10.78226487697149, −10.36938336469393, −9.728111927885424, −9.380892828158035, −8.808528859706879, −8.101254988689406, −7.885606194516453, −7.249170185203729, −6.570577186507626, −6.202952209320019, −5.504480798127532, −5.168622029097055, −4.385317831529336, −3.808482994317830, −3.343001254758334, −2.511872310910254, −2.098309506625961, −1.294415804415366, −0.2271961847648769,
0.2271961847648769, 1.294415804415366, 2.098309506625961, 2.511872310910254, 3.343001254758334, 3.808482994317830, 4.385317831529336, 5.168622029097055, 5.504480798127532, 6.202952209320019, 6.570577186507626, 7.249170185203729, 7.885606194516453, 8.101254988689406, 8.808528859706879, 9.380892828158035, 9.728111927885424, 10.36938336469393, 10.78226487697149, 11.37746999212027, 11.63221975411945, 12.34733760645245, 12.92694288683306, 13.31612034515179, 13.58461690049298
