| L(s) = 1 | − 5-s − 11-s − 2·13-s + 2·17-s + 4·19-s − 4·23-s + 25-s + 6·29-s + 2·37-s + 6·41-s − 12·43-s − 8·47-s + 6·53-s + 55-s − 8·59-s + 14·61-s + 2·65-s − 12·67-s + 8·71-s − 10·73-s − 4·79-s − 16·83-s − 2·85-s + 10·89-s − 4·95-s − 6·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.328·37-s + 0.937·41-s − 1.82·43-s − 1.16·47-s + 0.824·53-s + 0.134·55-s − 1.04·59-s + 1.79·61-s + 0.248·65-s − 1.46·67-s + 0.949·71-s − 1.17·73-s − 0.450·79-s − 1.75·83-s − 0.216·85-s + 1.05·89-s − 0.410·95-s − 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.236833734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.236833734\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.26823218031555, −12.01706446374971, −11.65182071543068, −11.21823166419556, −10.60368312486213, −10.11325841458537, −9.836717815575172, −9.435368880957797, −8.700916456989021, −8.286547192443008, −7.987072984642447, −7.338255339491172, −7.131060395224675, −6.463498239546875, −5.957465596699735, −5.468269804525647, −4.891120929232292, −4.583556008760539, −3.908345103127203, −3.409626961959748, −2.865961422791907, −2.444752713492316, −1.622547062114845, −1.100368033252773, −0.3041833664016500,
0.3041833664016500, 1.100368033252773, 1.622547062114845, 2.444752713492316, 2.865961422791907, 3.409626961959748, 3.908345103127203, 4.583556008760539, 4.891120929232292, 5.468269804525647, 5.957465596699735, 6.463498239546875, 7.131060395224675, 7.338255339491172, 7.987072984642447, 8.286547192443008, 8.700916456989021, 9.435368880957797, 9.836717815575172, 10.11325841458537, 10.60368312486213, 11.21823166419556, 11.65182071543068, 12.01706446374971, 12.26823218031555
