| L(s) = 1 | + 1.73·5-s + 7-s + 3.46·11-s + 5·13-s + 1.73·17-s + 2·19-s − 3.46·23-s − 2.00·25-s + 1.73·29-s + 2·31-s + 1.73·35-s − 7·37-s − 6.92·41-s + 8·43-s + 10.3·47-s + 49-s − 13.8·53-s + 5.99·55-s − 10.3·59-s − 61-s + 8.66·65-s + 2·67-s + 11·73-s + 3.46·77-s + 14·79-s − 3.46·83-s + 2.99·85-s + ⋯ |
| L(s) = 1 | + 0.774·5-s + 0.377·7-s + 1.04·11-s + 1.38·13-s + 0.420·17-s + 0.458·19-s − 0.722·23-s − 0.400·25-s + 0.321·29-s + 0.359·31-s + 0.292·35-s − 1.15·37-s − 1.08·41-s + 1.21·43-s + 1.51·47-s + 0.142·49-s − 1.90·53-s + 0.809·55-s − 1.35·59-s − 0.128·61-s + 1.07·65-s + 0.244·67-s + 1.28·73-s + 0.394·77-s + 1.57·79-s − 0.380·83-s + 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.532027332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.532027332\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 + 1.73T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.072816093438087714260956216902, −8.338441837433368339445617442390, −7.53231763980010663747111648373, −6.44270235831495573613343682420, −6.03611429095438001809706493767, −5.15057417688175895617129518941, −4.07977990615109222406238488382, −3.32567588769675931081988500314, −1.96091800486394431434211509337, −1.15835922364910855749890096371,
1.15835922364910855749890096371, 1.96091800486394431434211509337, 3.32567588769675931081988500314, 4.07977990615109222406238488382, 5.15057417688175895617129518941, 6.03611429095438001809706493767, 6.44270235831495573613343682420, 7.53231763980010663747111648373, 8.338441837433368339445617442390, 9.072816093438087714260956216902
