| L(s) = 1 | − 4·5-s + 2·7-s − 3·9-s + 6·11-s + 13-s + 17-s + 8·19-s − 4·23-s + 11·25-s + 6·29-s + 2·31-s − 8·35-s + 8·37-s + 4·43-s + 12·45-s − 3·49-s + 6·53-s − 24·55-s + 10·61-s − 6·63-s − 4·65-s − 8·67-s − 2·71-s + 12·77-s + 9·81-s − 4·83-s − 4·85-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.755·7-s − 9-s + 1.80·11-s + 0.277·13-s + 0.242·17-s + 1.83·19-s − 0.834·23-s + 11/5·25-s + 1.11·29-s + 0.359·31-s − 1.35·35-s + 1.31·37-s + 0.609·43-s + 1.78·45-s − 3/7·49-s + 0.824·53-s − 3.23·55-s + 1.28·61-s − 0.755·63-s − 0.496·65-s − 0.977·67-s − 0.237·71-s + 1.36·77-s + 81-s − 0.439·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.946817894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.946817894\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−16.14566230810453, −15.65045807419029, −14.90848082627627, −14.44271933123477, −14.18952383670600, −13.50358616544104, −12.42915052401631, −11.85141534597140, −11.70674936195106, −11.37921561037038, −10.67237926748277, −9.723298298294834, −9.115463728700530, −8.429802701508057, −8.091343674333895, −7.493786521058440, −6.847384373804823, −6.113975092784832, −5.349801094424065, −4.507385332355536, −4.024633280117341, −3.399959476777765, −2.717880032313450, −1.306732780365540, −0.7127304666266617,
0.7127304666266617, 1.306732780365540, 2.717880032313450, 3.399959476777765, 4.024633280117341, 4.507385332355536, 5.349801094424065, 6.113975092784832, 6.847384373804823, 7.493786521058440, 8.091343674333895, 8.429802701508057, 9.115463728700530, 9.723298298294834, 10.67237926748277, 11.37921561037038, 11.70674936195106, 11.85141534597140, 12.42915052401631, 13.50358616544104, 14.18952383670600, 14.44271933123477, 14.90848082627627, 15.65045807419029, 16.14566230810453
