| L(s) = 1 | + 5-s − 2·7-s − 11-s − 4·13-s + 4·19-s + 4·23-s + 25-s − 6·29-s + 8·31-s − 2·35-s − 10·37-s + 6·41-s − 6·43-s + 12·47-s − 3·49-s + 10·53-s − 55-s + 4·59-s − 14·61-s − 4·65-s + 8·67-s − 8·71-s − 4·73-s + 2·77-s − 14·83-s + 18·89-s + 8·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.301·11-s − 1.10·13-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.338·35-s − 1.64·37-s + 0.937·41-s − 0.914·43-s + 1.75·47-s − 3/7·49-s + 1.37·53-s − 0.134·55-s + 0.520·59-s − 1.79·61-s − 0.496·65-s + 0.977·67-s − 0.949·71-s − 0.468·73-s + 0.227·77-s − 1.53·83-s + 1.90·89-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 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
−7.26553199185379668057175357479, −6.99889957121846817641079851245, −6.05793234094430033116909948842, −5.40006200920121405736145086456, −4.83005187631745001202755198932, −3.83956506729129337456646656981, −2.96446253460953418420161520767, −2.41786640438621265719329071115, −1.23698159378796321812755169808, 0,
1.23698159378796321812755169808, 2.41786640438621265719329071115, 2.96446253460953418420161520767, 3.83956506729129337456646656981, 4.83005187631745001202755198932, 5.40006200920121405736145086456, 6.05793234094430033116909948842, 6.99889957121846817641079851245, 7.26553199185379668057175357479
