| L(s) = 1 | + 3.30·3-s − 0.302·5-s + 7.90·9-s − 2.60·11-s − 0.605·13-s − 1.00·15-s + 17-s + 6·19-s + 4.60·23-s − 4.90·25-s + 16.2·27-s − 1.39·29-s + 10.3·31-s − 8.60·33-s − 7.21·37-s − 2.00·39-s + 8.51·41-s − 0.697·43-s − 2.39·45-s − 10·47-s + 3.30·51-s + 4.30·53-s + 0.788·55-s + 19.8·57-s + 9.21·59-s + 0.697·61-s + 0.183·65-s + ⋯ |
| L(s) = 1 | + 1.90·3-s − 0.135·5-s + 2.63·9-s − 0.785·11-s − 0.167·13-s − 0.258·15-s + 0.242·17-s + 1.37·19-s + 0.960·23-s − 0.981·25-s + 3.11·27-s − 0.258·29-s + 1.85·31-s − 1.49·33-s − 1.18·37-s − 0.320·39-s + 1.32·41-s − 0.106·43-s − 0.356·45-s − 1.45·47-s + 0.462·51-s + 0.591·53-s + 0.106·55-s + 2.62·57-s + 1.19·59-s + 0.0892·61-s + 0.0227·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.949090582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.949090582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 + 0.302T + 5T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 + 0.605T + 13T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 4.60T + 23T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 7.21T + 37T^{2} \) |
| 41 | \( 1 - 8.51T + 41T^{2} \) |
| 43 | \( 1 + 0.697T + 43T^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 - 4.30T + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 - 0.697T + 61T^{2} \) |
| 67 | \( 1 - 6.30T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 4.51T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 5.21T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.506688500161231004311711585035, −7.961124940640164801546031569083, −7.43376244387374215714960849448, −6.73764853677029265620042500565, −5.42672818410414175540176096598, −4.60398173680048519162805186593, −3.66158674754342727022811804463, −2.99898620483043452236770836242, −2.32490400693043135370915291419, −1.17417811731172735545052185636,
1.17417811731172735545052185636, 2.32490400693043135370915291419, 2.99898620483043452236770836242, 3.66158674754342727022811804463, 4.60398173680048519162805186593, 5.42672818410414175540176096598, 6.73764853677029265620042500565, 7.43376244387374215714960849448, 7.961124940640164801546031569083, 8.506688500161231004311711585035
