| L(s) = 1 | − 5·5-s + 6·7-s + 9·9-s + 18·11-s + 6·13-s + 2·19-s − 26·23-s + 25·25-s − 30·35-s + 54·37-s − 78·41-s − 45·45-s + 86·47-s − 13·49-s − 74·53-s − 90·55-s − 78·59-s + 54·63-s − 30·65-s + 108·77-s + 81·81-s + 18·89-s + 36·91-s − 10·95-s + 162·99-s − 186·103-s + 130·115-s + ⋯ |
| L(s) = 1 | − 5-s + 6/7·7-s + 9-s + 1.63·11-s + 6/13·13-s + 2/19·19-s − 1.13·23-s + 25-s − 6/7·35-s + 1.45·37-s − 1.90·41-s − 45-s + 1.82·47-s − 0.265·49-s − 1.39·53-s − 1.63·55-s − 1.32·59-s + 6/7·63-s − 0.461·65-s + 1.40·77-s + 81-s + 0.202·89-s + 0.395·91-s − 0.105·95-s + 1.63·99-s − 1.80·103-s + 1.13·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.530645798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.530645798\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 - 6 T + p^{2} T^{2} \) |
| 11 | \( 1 - 18 T + p^{2} T^{2} \) |
| 13 | \( 1 - 6 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 - 2 T + p^{2} T^{2} \) |
| 23 | \( 1 + 26 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 54 T + p^{2} T^{2} \) |
| 41 | \( 1 + 78 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( 1 - 86 T + p^{2} T^{2} \) |
| 53 | \( 1 + 74 T + p^{2} T^{2} \) |
| 59 | \( 1 + 78 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 - 18 T + p^{2} T^{2} \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−12.37467690027803987385963976127, −11.72802669660607792407034950744, −10.85282859431548997997744806467, −9.550815551971685796162487854010, −8.417018213889008588201055769722, −7.48074721113275939316174974911, −6.37901802451116508350278906841, −4.57787252540325384887507575134, −3.78159770451926161841936535557, −1.40170264901865770549568521218,
1.40170264901865770549568521218, 3.78159770451926161841936535557, 4.57787252540325384887507575134, 6.37901802451116508350278906841, 7.48074721113275939316174974911, 8.417018213889008588201055769722, 9.550815551971685796162487854010, 10.85282859431548997997744806467, 11.72802669660607792407034950744, 12.37467690027803987385963976127
