| L(s) = 1 | + 2.70·3-s + 3.77·7-s + 4.31·9-s + 4.71·11-s − 5.44·13-s − 2.40·17-s + 19-s + 10.2·21-s + 4.38·23-s + 3.55·27-s + 9.05·29-s − 9.92·31-s + 12.7·33-s + 10.0·37-s − 14.7·39-s + 1.11·41-s − 12.1·43-s + 6.72·47-s + 7.28·49-s − 6.51·51-s + 4.34·53-s + 2.70·57-s − 4.03·59-s − 9.14·61-s + 16.3·63-s + 4.39·67-s + 11.8·69-s + ⋯ |
| L(s) = 1 | + 1.56·3-s + 1.42·7-s + 1.43·9-s + 1.42·11-s − 1.50·13-s − 0.584·17-s + 0.229·19-s + 2.23·21-s + 0.915·23-s + 0.684·27-s + 1.68·29-s − 1.78·31-s + 2.22·33-s + 1.65·37-s − 2.35·39-s + 0.174·41-s − 1.84·43-s + 0.980·47-s + 1.04·49-s − 0.912·51-s + 0.597·53-s + 0.358·57-s − 0.525·59-s − 1.17·61-s + 2.05·63-s + 0.537·67-s + 1.42·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.265256387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.265256387\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.70T + 3T^{2} \) |
| 7 | \( 1 - 3.77T + 7T^{2} \) |
| 11 | \( 1 - 4.71T + 11T^{2} \) |
| 13 | \( 1 + 5.44T + 13T^{2} \) |
| 17 | \( 1 + 2.40T + 17T^{2} \) |
| 23 | \( 1 - 4.38T + 23T^{2} \) |
| 29 | \( 1 - 9.05T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 1.11T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 - 6.72T + 47T^{2} \) |
| 53 | \( 1 - 4.34T + 53T^{2} \) |
| 59 | \( 1 + 4.03T + 59T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 - 4.39T + 67T^{2} \) |
| 71 | \( 1 - 7.90T + 71T^{2} \) |
| 73 | \( 1 - 3.07T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 4.37T + 83T^{2} \) |
| 89 | \( 1 - 2.53T + 89T^{2} \) |
| 97 | \( 1 + 0.302T + 97T^{2} \) |
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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
−8.460520615299712274801513924869, −7.900928175317174353723216912910, −7.24149927553213894423313854070, −6.59952482346512916830842356875, −5.21180014944874551341832886002, −4.55827124859823565158939970972, −3.87591256873129994719364903588, −2.83285418013584139551665376052, −2.10314621713100250303358454587, −1.25483300358077305626455926866,
1.25483300358077305626455926866, 2.10314621713100250303358454587, 2.83285418013584139551665376052, 3.87591256873129994719364903588, 4.55827124859823565158939970972, 5.21180014944874551341832886002, 6.59952482346512916830842356875, 7.24149927553213894423313854070, 7.900928175317174353723216912910, 8.460520615299712274801513924869
