L(s) = 1 | + 3.12·3-s − 5-s − 7-s + 6.76·9-s + 5.76·11-s + 1.18·13-s − 3.12·15-s − 7.93·17-s − 0.642·19-s − 3.12·21-s + 7.81·23-s + 25-s + 11.7·27-s + 9.81·29-s − 8.66·31-s + 18.0·33-s + 35-s − 6.80·37-s + 3.71·39-s + 7.47·41-s − 43-s − 6.76·45-s + 12.6·47-s + 49-s − 24.7·51-s − 8.21·53-s − 5.76·55-s + ⋯ |
L(s) = 1 | + 1.80·3-s − 0.447·5-s − 0.377·7-s + 2.25·9-s + 1.73·11-s + 0.329·13-s − 0.806·15-s − 1.92·17-s − 0.147·19-s − 0.681·21-s + 1.62·23-s + 0.200·25-s + 2.26·27-s + 1.82·29-s − 1.55·31-s + 3.13·33-s + 0.169·35-s − 1.11·37-s + 0.594·39-s + 1.16·41-s − 0.152·43-s − 1.00·45-s + 1.84·47-s + 0.142·49-s − 3.47·51-s − 1.12·53-s − 0.777·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.139103577\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.139103577\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 - 3.12T + 3T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 - 1.18T + 13T^{2} \) |
| 17 | \( 1 + 7.93T + 17T^{2} \) |
| 19 | \( 1 + 0.642T + 19T^{2} \) |
| 23 | \( 1 - 7.81T + 23T^{2} \) |
| 29 | \( 1 - 9.81T + 29T^{2} \) |
| 31 | \( 1 + 8.66T + 31T^{2} \) |
| 37 | \( 1 + 6.80T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 8.21T + 53T^{2} \) |
| 59 | \( 1 - 0.705T + 59T^{2} \) |
| 61 | \( 1 + 4.03T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 4.05T + 71T^{2} \) |
| 73 | \( 1 - 4.31T + 73T^{2} \) |
| 79 | \( 1 - 5.17T + 79T^{2} \) |
| 83 | \( 1 - 6.02T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 5.87T + 97T^{2} \) |
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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
−8.361995713259788795236297939836, −7.28944822052882899370370256869, −6.89822682132430768840314488221, −6.32932756688397254105486155082, −4.83278749025996076884557400664, −4.11297248916598961550697372561, −3.65143756321169627397142091068, −2.85278034584286698614528509077, −2.03686895084713465009380403961, −1.03878190212523688362476362706,
1.03878190212523688362476362706, 2.03686895084713465009380403961, 2.85278034584286698614528509077, 3.65143756321169627397142091068, 4.11297248916598961550697372561, 4.83278749025996076884557400664, 6.32932756688397254105486155082, 6.89822682132430768840314488221, 7.28944822052882899370370256869, 8.361995713259788795236297939836