| L(s) = 1 | − 3·7-s + 3·11-s − 13-s + 5·17-s + 6·19-s − 3·23-s − 2·29-s − 6·31-s + 3·37-s + 5·41-s + 2·49-s − 7·53-s + 61-s + 15·71-s − 2·73-s − 9·77-s + 15·79-s − 6·83-s + 11·89-s + 3·91-s + 97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 15·119-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.904·11-s − 0.277·13-s + 1.21·17-s + 1.37·19-s − 0.625·23-s − 0.371·29-s − 1.07·31-s + 0.493·37-s + 0.780·41-s + 2/7·49-s − 0.961·53-s + 0.128·61-s + 1.78·71-s − 0.234·73-s − 1.02·77-s + 1.68·79-s − 0.658·83-s + 1.16·89-s + 0.314·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.37·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.132199811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.132199811\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.85059694906530, −13.37899922631093, −12.70275352080981, −12.38406478981624, −11.98358735931060, −11.41352926494351, −10.90164894066046, −10.26164015873837, −9.619943493104569, −9.474986721607156, −9.152326756521894, −8.144364144222557, −7.814648109936376, −7.221879472123613, −6.702499078939326, −6.204832657149703, −5.587354539506958, −5.255509559881488, −4.361496728385851, −3.711104256186066, −3.379411258287351, −2.797485560870213, −1.948799223102460, −1.196677552142617, −0.5032790542877308,
0.5032790542877308, 1.196677552142617, 1.948799223102460, 2.797485560870213, 3.379411258287351, 3.711104256186066, 4.361496728385851, 5.255509559881488, 5.587354539506958, 6.204832657149703, 6.702499078939326, 7.221879472123613, 7.814648109936376, 8.144364144222557, 9.152326756521894, 9.474986721607156, 9.619943493104569, 10.26164015873837, 10.90164894066046, 11.41352926494351, 11.98358735931060, 12.38406478981624, 12.70275352080981, 13.37899922631093, 13.85059694906530
