| L(s) = 1 | − 7-s − 3·9-s − 6·11-s − 13-s − 4·17-s + 5·19-s + 3·23-s + 5·29-s + 3·31-s − 4·37-s − 6·41-s + 43-s + 7·47-s + 49-s − 9·53-s + 8·59-s + 10·61-s + 3·63-s + 6·67-s + 8·71-s + 13·73-s + 6·77-s − 3·79-s + 9·81-s − 15·83-s + 3·89-s + 91-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 1.80·11-s − 0.277·13-s − 0.970·17-s + 1.14·19-s + 0.625·23-s + 0.928·29-s + 0.538·31-s − 0.657·37-s − 0.937·41-s + 0.152·43-s + 1.02·47-s + 1/7·49-s − 1.23·53-s + 1.04·59-s + 1.28·61-s + 0.377·63-s + 0.733·67-s + 0.949·71-s + 1.52·73-s + 0.683·77-s − 0.337·79-s + 81-s − 1.64·83-s + 0.317·89-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.055737409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.055737409\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.40260756716131, −12.86634454875650, −12.52793201050203, −11.93227870116027, −11.32853126776836, −11.11435690138053, −10.40638036235834, −10.11312038786492, −9.555819250320449, −8.985265513721771, −8.397433367371964, −8.165144286034956, −7.533867353168478, −6.925637186597319, −6.590325800212400, −5.794427590709790, −5.339410676298217, −5.022429548881360, −4.447675864328952, −3.483588875673267, −3.141588946716367, −2.441148805767509, −2.257476989649898, −1.020816373963159, −0.3398265215476559,
0.3398265215476559, 1.020816373963159, 2.257476989649898, 2.441148805767509, 3.141588946716367, 3.483588875673267, 4.447675864328952, 5.022429548881360, 5.339410676298217, 5.794427590709790, 6.590325800212400, 6.925637186597319, 7.533867353168478, 8.165144286034956, 8.397433367371964, 8.985265513721771, 9.555819250320449, 10.11312038786492, 10.40638036235834, 11.11435690138053, 11.32853126776836, 11.93227870116027, 12.52793201050203, 12.86634454875650, 13.40260756716131
