| L(s) = 1 | − 3·5-s + 11-s + 13-s − 2·17-s + 7·19-s − 4·23-s + 4·25-s − 7·29-s + 2·31-s + 37-s − 12·41-s + 12·43-s + 9·47-s − 12·53-s − 3·55-s − 3·59-s − 10·61-s − 3·65-s + 7·67-s − 12·71-s + 15·73-s − 12·79-s + 18·83-s + 6·85-s − 10·89-s − 21·95-s + 4·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.301·11-s + 0.277·13-s − 0.485·17-s + 1.60·19-s − 0.834·23-s + 4/5·25-s − 1.29·29-s + 0.359·31-s + 0.164·37-s − 1.87·41-s + 1.82·43-s + 1.31·47-s − 1.64·53-s − 0.404·55-s − 0.390·59-s − 1.28·61-s − 0.372·65-s + 0.855·67-s − 1.42·71-s + 1.75·73-s − 1.35·79-s + 1.97·83-s + 0.650·85-s − 1.05·89-s − 2.15·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.110684563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.110684563\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.95150678123819, −13.67581225942860, −13.00068804689689, −12.29955023626834, −12.06345879950653, −11.61258432237195, −10.98704639418399, −10.83173781481133, −9.918944927198410, −9.456499492745124, −8.983662202461916, −8.359005848409483, −7.772401943089481, −7.537002565811276, −6.966606391952744, −6.293331127836713, −5.704055116226000, −5.101894634423618, −4.395186637107594, −3.960502799037230, −3.423580210362432, −2.896221515534415, −1.965510607239287, −1.215666513527892, −0.3717650940060657,
0.3717650940060657, 1.215666513527892, 1.965510607239287, 2.896221515534415, 3.423580210362432, 3.960502799037230, 4.395186637107594, 5.101894634423618, 5.704055116226000, 6.293331127836713, 6.966606391952744, 7.537002565811276, 7.772401943089481, 8.359005848409483, 8.983662202461916, 9.456499492745124, 9.918944927198410, 10.83173781481133, 10.98704639418399, 11.61258432237195, 12.06345879950653, 12.29955023626834, 13.00068804689689, 13.67581225942860, 13.95150678123819
