| L(s) = 1 | + 16·4-s + 71·7-s − 337·13-s + 256·16-s − 601·19-s + 625·25-s + 1.13e3·28-s + 194·31-s − 529·37-s − 3.21e3·43-s + 2.64e3·49-s − 5.39e3·52-s + 7.19e3·61-s + 4.09e3·64-s + 2.90e3·67-s − 1.24e3·73-s − 9.61e3·76-s + 4.67e3·79-s − 2.39e4·91-s + 9.07e3·97-s + 1.00e4·100-s − 1.98e4·103-s + 2.20e4·109-s + 1.81e4·112-s + ⋯ |
| L(s) = 1 | + 4-s + 1.44·7-s − 1.99·13-s + 16-s − 1.66·19-s + 25-s + 1.44·28-s + 0.201·31-s − 0.386·37-s − 1.73·43-s + 1.09·49-s − 1.99·52-s + 1.93·61-s + 64-s + 0.646·67-s − 0.234·73-s − 1.66·76-s + 0.749·79-s − 2.88·91-s + 0.964·97-s + 100-s − 1.87·103-s + 1.85·109-s + 1.44·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.667562511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.667562511\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 7 | \( 1 - 71 T + p^{4} T^{2} \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 + 337 T + p^{4} T^{2} \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( 1 + 601 T + p^{4} T^{2} \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 - 194 T + p^{4} T^{2} \) |
| 37 | \( 1 + 529 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 + 3214 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 - 7199 T + p^{4} T^{2} \) |
| 67 | \( 1 - 2903 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 + 1249 T + p^{4} T^{2} \) |
| 79 | \( 1 - 4679 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( 1 - 9071 T + p^{4} T^{2} \) |
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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
−16.75263768577637040989235941169, −15.04952294764606219841084010105, −14.56756900868376874302879694925, −12.51074638815196116965650809016, −11.44866451912438386718790765007, −10.29776587991541655436527843802, −8.238964733148844902725521644900, −6.95123716708336977906275478123, −4.94211312981032595816400801936, −2.17012470140008254428436281074,
2.17012470140008254428436281074, 4.94211312981032595816400801936, 6.95123716708336977906275478123, 8.238964733148844902725521644900, 10.29776587991541655436527843802, 11.44866451912438386718790765007, 12.51074638815196116965650809016, 14.56756900868376874302879694925, 15.04952294764606219841084010105, 16.75263768577637040989235941169
