| L(s) = 1 | + 5.55e4·3-s + 1.95e6·5-s + 1.74e8·7-s + 1.92e9·9-s − 1.00e10·11-s + 4.72e10·13-s + 1.08e11·15-s + 3.78e11·17-s + 8.71e11·19-s + 9.68e12·21-s − 1.24e13·23-s + 3.81e12·25-s + 4.25e13·27-s − 8.06e13·29-s − 1.30e14·31-s − 5.60e14·33-s + 3.40e14·35-s + 1.32e15·37-s + 2.62e15·39-s − 1.01e15·41-s + 2.64e15·43-s + 3.76e15·45-s − 1.10e16·47-s + 1.89e16·49-s + 2.10e16·51-s − 1.60e16·53-s − 1.96e16·55-s + ⋯ |
| L(s) = 1 | + 1.63·3-s + 0.447·5-s + 1.63·7-s + 1.65·9-s − 1.28·11-s + 1.23·13-s + 0.729·15-s + 0.773·17-s + 0.619·19-s + 2.66·21-s − 1.43·23-s + 0.199·25-s + 1.07·27-s − 1.03·29-s − 0.888·31-s − 2.10·33-s + 0.729·35-s + 1.68·37-s + 2.01·39-s − 0.482·41-s + 0.802·43-s + 0.741·45-s − 1.43·47-s + 1.66·49-s + 1.26·51-s − 0.669·53-s − 0.576·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(4.960627788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.960627788\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 1.95e6T \) |
| good | 3 | \( 1 - 5.55e4T + 1.16e9T^{2} \) |
| 7 | \( 1 - 1.74e8T + 1.13e16T^{2} \) |
| 11 | \( 1 + 1.00e10T + 6.11e19T^{2} \) |
| 13 | \( 1 - 4.72e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 3.78e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 8.71e11T + 1.97e24T^{2} \) |
| 23 | \( 1 + 1.24e13T + 7.46e25T^{2} \) |
| 29 | \( 1 + 8.06e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 1.30e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.32e15T + 6.24e29T^{2} \) |
| 41 | \( 1 + 1.01e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 2.64e15T + 1.08e31T^{2} \) |
| 47 | \( 1 + 1.10e16T + 5.88e31T^{2} \) |
| 53 | \( 1 + 1.60e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 3.99e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.20e17T + 8.34e33T^{2} \) |
| 67 | \( 1 - 2.46e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 6.01e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 2.96e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 5.37e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.33e18T + 2.90e36T^{2} \) |
| 89 | \( 1 - 2.20e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 1.28e19T + 5.60e37T^{2} \) |
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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
−14.04806438867155030458906554957, −13.08850889215652841592898454959, −11.12778807091240450648783797787, −9.699434474580443021644797513050, −8.250082155572741333000320311902, −7.79167354065531729151865683967, −5.41549114136819172486396331738, −3.80331685909603356199092447847, −2.36273977354261506873796612767, −1.41617163529332659059103279016,
1.41617163529332659059103279016, 2.36273977354261506873796612767, 3.80331685909603356199092447847, 5.41549114136819172486396331738, 7.79167354065531729151865683967, 8.250082155572741333000320311902, 9.699434474580443021644797513050, 11.12778807091240450648783797787, 13.08850889215652841592898454959, 14.04806438867155030458906554957
