| L(s) = 1 | + 2·5-s + 2·7-s + 4·11-s + 2·13-s + 4·17-s − 4·19-s − 8·23-s − 25-s + 6·29-s − 6·31-s + 4·35-s − 2·37-s + 12·41-s + 12·43-s − 8·47-s − 3·49-s + 6·53-s + 8·55-s + 8·59-s − 10·61-s + 4·65-s − 8·67-s + 2·73-s + 8·77-s − 14·79-s + 12·83-s + 8·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 1.20·11-s + 0.554·13-s + 0.970·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.07·31-s + 0.676·35-s − 0.328·37-s + 1.87·41-s + 1.82·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s + 1.07·55-s + 1.04·59-s − 1.28·61-s + 0.496·65-s − 0.977·67-s + 0.234·73-s + 0.911·77-s − 1.57·79-s + 1.31·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.256534198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.256534198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.782683048374999514747740510300, −9.017830298727617035965492530881, −8.239699972575940949478402632684, −7.36183700646228938196129724807, −6.11969843230623172416001421015, −5.87618764591883614635582577159, −4.54054370996283917506505788819, −3.72962610687736412807879856876, −2.22088860133337487727877949530, −1.31006766181093226223763545704,
1.31006766181093226223763545704, 2.22088860133337487727877949530, 3.72962610687736412807879856876, 4.54054370996283917506505788819, 5.87618764591883614635582577159, 6.11969843230623172416001421015, 7.36183700646228938196129724807, 8.239699972575940949478402632684, 9.017830298727617035965492530881, 9.782683048374999514747740510300
