| L(s) = 1 | + 5-s − 7-s − 3·9-s + 4·11-s − 2·13-s + 6·17-s − 4·19-s + 8·23-s + 25-s − 6·29-s − 35-s − 10·37-s + 10·41-s + 12·43-s − 3·45-s + 49-s + 6·53-s + 4·55-s + 12·59-s − 2·61-s + 3·63-s − 2·65-s + 4·67-s + 12·71-s − 10·73-s − 4·77-s + 4·79-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 9-s + 1.20·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.169·35-s − 1.64·37-s + 1.56·41-s + 1.82·43-s − 0.447·45-s + 1/7·49-s + 0.824·53-s + 0.539·55-s + 1.56·59-s − 0.256·61-s + 0.377·63-s − 0.248·65-s + 0.488·67-s + 1.42·71-s − 1.17·73-s − 0.455·77-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.823392542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.823392542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.187225597375106708489731721786, −8.417589260818054771104729716468, −7.37243974974671682839526925945, −6.71587041632029966830258649128, −5.79311744359521732356727701248, −5.30380928242354662070686082447, −4.05557494890148995121270932880, −3.21199274145712931199180881237, −2.24903032243261477802323164785, −0.892598199057904506325797054206,
0.892598199057904506325797054206, 2.24903032243261477802323164785, 3.21199274145712931199180881237, 4.05557494890148995121270932880, 5.30380928242354662070686082447, 5.79311744359521732356727701248, 6.71587041632029966830258649128, 7.37243974974671682839526925945, 8.417589260818054771104729716468, 9.187225597375106708489731721786
