| L(s) = 1 | + 7·7-s + 9·9-s + 6·11-s − 18·23-s + 25·25-s − 54·29-s − 38·37-s − 58·43-s + 49·49-s − 6·53-s + 63·63-s + 118·67-s − 114·71-s + 42·77-s + 94·79-s + 81·81-s + 54·99-s − 186·107-s + 106·109-s − 222·113-s + ⋯ |
| L(s) = 1 | + 7-s + 9-s + 6/11·11-s − 0.782·23-s + 25-s − 1.86·29-s − 1.02·37-s − 1.34·43-s + 49-s − 0.113·53-s + 63-s + 1.76·67-s − 1.60·71-s + 6/11·77-s + 1.18·79-s + 81-s + 6/11·99-s − 1.73·107-s + 0.972·109-s − 1.96·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.552790139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.552790139\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 - 6 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 + 18 T + p^{2} T^{2} \) |
| 29 | \( 1 + 54 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 38 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 58 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 + 6 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 - 118 T + p^{2} T^{2} \) |
| 71 | \( 1 + 114 T + p^{2} T^{2} \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 - 94 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.39122471308763858267294405234, −12.31624646632809714584721904101, −11.31546826874976778009866223217, −10.27370473884268084195628427706, −9.085539636300195382565728215471, −7.87653073986229305688657160001, −6.79348099700076131656558260201, −5.19307015109789144408536576153, −3.93443463208281217298642402566, −1.68325247529620951053373397458,
1.68325247529620951053373397458, 3.93443463208281217298642402566, 5.19307015109789144408536576153, 6.79348099700076131656558260201, 7.87653073986229305688657160001, 9.085539636300195382565728215471, 10.27370473884268084195628427706, 11.31546826874976778009866223217, 12.31624646632809714584721904101, 13.39122471308763858267294405234
