| L(s) = 1 | − 2.95·3-s − 5-s − 7-s + 5.73·9-s + 2.16·11-s − 5.99·13-s + 2.95·15-s + 0.748·17-s + 7.21·19-s + 2.95·21-s + 1.78·23-s + 25-s − 8.07·27-s + 6.35·29-s − 7.01·31-s − 6.38·33-s + 35-s − 10.6·37-s + 17.7·39-s + 9.69·41-s − 43-s − 5.73·45-s − 10.9·47-s + 49-s − 2.21·51-s + 9.15·53-s − 2.16·55-s + ⋯ |
| L(s) = 1 | − 1.70·3-s − 0.447·5-s − 0.377·7-s + 1.91·9-s + 0.651·11-s − 1.66·13-s + 0.762·15-s + 0.181·17-s + 1.65·19-s + 0.644·21-s + 0.372·23-s + 0.200·25-s − 1.55·27-s + 1.18·29-s − 1.25·31-s − 1.11·33-s + 0.169·35-s − 1.75·37-s + 2.83·39-s + 1.51·41-s − 0.152·43-s − 0.854·45-s − 1.60·47-s + 0.142·49-s − 0.309·51-s + 1.25·53-s − 0.291·55-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
| \(L(1)\) |
\(\approx\) |
\(0.6280874327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6280874327\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.95T + 3T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 13 | \( 1 + 5.99T + 13T^{2} \) |
| 17 | \( 1 - 0.748T + 17T^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 23 | \( 1 - 1.78T + 23T^{2} \) |
| 29 | \( 1 - 6.35T + 29T^{2} \) |
| 31 | \( 1 + 7.01T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 9.69T + 41T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 9.15T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 1.07T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 4.27T + 71T^{2} \) |
| 73 | \( 1 - 2.34T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 1.94T + 83T^{2} \) |
| 89 | \( 1 - 0.189T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−7.72989417909773411764858718414, −7.12902188997654947535262825532, −6.77452709171381657415485488282, −5.84555228360355583061083365924, −5.18057088589579795377319290389, −4.75905893699860646646513659645, −3.80727290282512973078909749448, −2.87884137268277528479164256985, −1.47886791482628912955310809291, −0.47997080924779371147009328854,
0.47997080924779371147009328854, 1.47886791482628912955310809291, 2.87884137268277528479164256985, 3.80727290282512973078909749448, 4.75905893699860646646513659645, 5.18057088589579795377319290389, 5.84555228360355583061083365924, 6.77452709171381657415485488282, 7.12902188997654947535262825532, 7.72989417909773411764858718414
